Tabellen zur Fourier Transformation

HERAUSGEGEBEN VON
R. GRAMMEL . E. HOPF . H. HOPF . F. K. SCHMIDT B. L. VAN DER WAERDEN
BAND XC
TABELLEN ZUR
AMERICAN UNIVERSITY WASHINGTON, D. C.
SPRINGER-VERLAG BERLIN· GOTTINGEN . HEIDELBERG
VORBEHALTEN
OHNE AUSDRUCKLICHE GENEHMIGUNG DES VERLAGES 1ST ES AUCH NICHT GESTATTET, DIESES BUCH ODER TEILE DARAUS
AUF PHOTHOMECHANISCHEM WEGE (PHOTOKOPIE, MIKROKOPIE) ZU VERVIELFALTIGEN
© BY SPRINGER-VERLAG OHG. BERLIN· GOTTINGEN . HEIDELBERG 1957 SOFfCOVER REPRINT OF THE HARDCOVER 1ST EDITION 1957
ISBN-13: 978-3-642-94701-8 DOl: 10.10071 978-3-642-94700-1
e-ISBN-13: 978-3-642-94700-1
MEINER FRAU DOROTHY
Die nachfolgenden Tabellen stellen eine Sammlung von Integralen der folgenden Form dar.
00
00
(2) g (y) = f I (x) sin (x y) d x (Zweites Kapitel) 0
00
(3 ) g(y) = Jt(x) eixy dx (Drittes Kapitel). -00
Die Funktion g(y) in (1), (2) und (3) wird der Reihe nach als FOURIER Kosinus-, FOURIER-Sinus-, und exponentielle FOURIER-Transformation der Funktion I (x) bezeichnet. Unter gewissen Bedingungen [s. z. B. eines der im Literaturverzeichnis unter a) aufgefiihrten WerkeJ gelten die (1), (2) und (3) entsprechenden Umkehrformeln
(1 a)
00
00
-00
Offensichtlich geht das Formelpaar (3), (3a) in (1), (1 a) oder (2), (2a) iiber, je nachdem I(x) gerade oder ungerade ist. In den Tabellen sind Parameter die durch lateinische Buchstaben bezeichnet sind, wenn nicht anders vermerkt, als positiv und reell vorausgesetzt, wobei fUr die Beispiele im dritten Kapitel der Parameter yauch negative Werte annimmt. In den meisten Fallen ist der Giiltigkeitsbereich eines Formel paares fUr komplexe Werte dieser GraBen sofort ersichtlich. Griechische Buchstaben bedeuten komplexe Parameter innerhalb des angegebenen Giiltigkeitsbereiches. In einigen Fallen ist die Funktion g (y) nur iiber einen Teilbereich von y angegeben. Dies bedeutet, daB sich g (y) fUr den restlichen Bereich nicht in einfacher Form angeben liiBt. Die historische Entwicklung der FOURIER-Transformation ist in dem Artikel
VIIT Vorwort
"Trigonometrische Reihen und Integrale" von H. BURKHARDT dar gestellt. Eine reichhaltige Auswahl von Losungen von Randwert problemen mittels der FOURIER-Transformation ist in dem Buche von SNEDDON gegeben. In diesem Zusammenhange sei bemerkt, daB in den letzten 10 Jahren die auf der FOURIER-Transformation beruhende Me thode von WIENER und HOPF zur Auflosung von singuHiren Integral gleichungen (s. z.B. Rap. 4 des Buches von PALEY und WIENER) in umfangreichem MaBe zur Losung von Randwertproblemen der Beu gungstheorie herangezogen wurde. Eine Reihe von Anwendungen in der Theorie der elektrischen Netzwerke findet sich in den I. R. E. Transactions on circuit theory". Die erste groBere Sammlung von FOURIER-Integralen scheint die Zusammenstellung von CAMPBELL im Bell System Technical Journal zu sein. Diese wurde spiHer von CAMP BELL und FOSTER erweitert und in Buchform herausgegeben. Weitere Tabellen enthalt das kiirzlich von A. ERDELYI herausgegebene Werk. Der vorliegende Band enthalt rund 1800 Formelpaare. Ein erheblicher Teil davon ist neu und ist unveroffentlichtem Material des Verfassers entnommen. Verbesserungsvorschlage, insbesondere die Richtigstellung von Irrtiimern und Druckfehlern werden vom Verfasser dankbar en t gegengenommen.
American University Washington, D.C. 1956
FRITZ OBERHETTINGER
Inhaltsverzeichnis Sei!e
Vorwort .................. . VII
§ 1. Algebraische Funktionen 1 § 2. Beliebige Potenzen . . . . 5 § 3. Exponentiaifunktionen . . 10 § 4. Logarithmische Funktionen 15 § 5. Trigonometrische Funktionen 18 § 6. Zyklometrische Funktionen 32 § 7. Hyperbolische Funktionen. . 34 § 8. Orthogonale Polynome . . . 43 § 9. Gamma- und RIEMANN-Zetafunktion 46
§ 10. Fehlerintegral . . . . . . . . . 47 § 11. Exponentialintegral. . . . . . . 49 § 12. Integralsinus und Integralkosinus . 50 § 13. FREsNEL-Integrale . . . . . . . 52 § 14. LEGENDRE-Funktionen . . . . . 55 § 15. BESsEL-Funktionen vom Argument x 61 § 16. BESsEL-Funktionen vom Argument x2 und 1/X 71 § 17. BESsEL-Funktionen vom Argument (ax2+bx+c)~ 74 § 18. BESSEL-Funktionen mit trigonometrischem und hyperbolischem
Argument . . . . . . . . . . . . . . . . . . 82 § 19. BESsEL-Funktionen mit variabler Ordnung. . . . . . . . 84 § 20. Modifizierte BESsEL-Funktionen vom Argument x 86 § 21. Modifizierte BESsEL-Funktionen vom Argument x2 und l/X 89
1 § 22. Modifizierte BESsEL-Funktionen vom Argument (ax·+ bx + c)" 91 § 23. Modifizierte BESsEL-Funktionen mit trigonometrischem und hyper-
bolischem Argument . . . . . . . . . . . . . . . 97 § 24. Modifizierte BESsEL-Funktionen mit variabler Ordnung 98 § 25. LOMMEL-Funktionen . . . . . 100 § 26. ANGER- und WEBER-Funktionen 101 § 27. STRuvE-Funktionen. . . . . . 103 § 28. Elliptische Integrale . . . . . 10 5 § 29. Parabolische Zylinderfunktionen 107 § 30. WHITTAKER-Funktionen . 108 § 31. Thetafunktionen . . . . . . . 109
Zweites Kapitel: FOURIER-Sinus-Transformationen
x Inhaltsverzeichnis Seite
§ 10. Fehlerintegral . . . . 157 § 11. Exponentialintegral. . 159 § 12. Integralsinus und Integralkosinus . 160 § 13. FREsNEL-Integrale . . . . . . . 162 § 14. LEGENDRE-Funktionen . . . . . 163 § 15. BESsEL-Funktionen vom Argument x 165 § 16. BESsEL-Funktionen vom Argument x2 und llx 174
§ 17. BESsEL-Funktionen vom Argument (ax2 + bx + c)~ 177 § 18. BESsEL-Funktionen mit trigonometrischem und hyperbolischem
Argnment . . . . . . . . . . . . . . . . . . 182 § 19. BESsEL-Funktionen mit variabler Ordnung. . . . 183 § 20. Modifizierte BESsEL-Funktionen vom Argument x 184 § 21. Modifizierte BESSEL-Funktionen vom Argument x 2 und 11x 186
§ 22. Modifizierte BESsEL-Funktionen vom Argument (ax2+ bx + c)~ 188 § 23. Modifizierte BESsEL-Funktionen mit trigonometrischem und hyper-
bolischem Argument . . . . . . . . . . . . . . . 191 § 24. Modifizierte BESsEL-Funktionen mit varia bIer Ordnung 192 § 25· LOMMEL-Funktionen . . . . . 194 § 26. ANGER- und WEBER-Funktionen 194 § 27· STRuvE-Funktionen. . . . . . 195 § 28. Elliptische Integrale . . . . . 197 § 29. Parabolische Zylinderfunktionen 198 § 30. WHITTAKER-Funktionen. . . . 199
Drittes Kapitel: Exponentielle FOURIER-Transformationen. 201
Anhang: Zusammenstellung von Abkiirzungen und Definitionen der Funk- tionssymbole 207
Literatur ..... .
Berichtigungen
S. 4, Formel 7, rechte Seite: (~ n Y' anstatt (+ n y Y . S. 10, Formel 7, rechte Seite: Faktor y fehlt.
S. 11, letzte Formel, linke Seite: x2n anstatt x~2n.
S. 14, Formel5, rechte Seite: (nb)~ anstatt nb~.
S. 21, Formel 4, linke Seite: + b2 anstatt - b2 im Nenner.
S. 36, Formel 6, rechte Seite: cos (~ 15) anstatt cos ( ~ b) im ~ellller.
S. 39, Formell, linke Seite: cosh[b(a2-x2)~J anstatt cosh [b(a2-x2)J.
S. 48, Formel6, linke Seite: -Erf(ax-1) anstatt Erfc(ax-1).
S. 49, Formel2, rechte Seite: Ei(-~ab+iay) anstatt i(-ab+iay).
S. 59, Formel 2, linke Seite: oberer Index f! fehlt.
214
1
o
x>2
n = 2,3,4, ...
4y-2cos y sin2 (~ y)
(2n)~ y-! [~ - C (a y) 1
- si (a y) sin (a y) - Ci (a y) cos (a y)
- sin (ay) si[y(a + b)] - cos(ay) X
xCi[y(a+b)]
X sin [~ n(n - m) 1 a-m (_ y)n-m-l
Oberhettinger, Tabellen
f(x) g(y) = if (x) cos (x y) dx o
0 O<xb + sin (ay) si(ay + by) - Ci(by)]
n-l L (m-1)!( +b)-m(_ )n-m-l (n-1)! a y X
m=l
0 O<x b 2 (n-1)!
n = 1,2,3, ... X [sin (ay + ~ nn) Ci(ay + by) -
- cos (ay + ~ nn)si(ay + by) 1
xl (a + xtl n!(2y)-~-na!cos(ay) [1- C(ay)-
-S (a y)]-nalsin(a y) [C(a y)-S (a y)]
x-lea + xtl na-l{cos(ay) [1 - C(ay) - S(ay)] +
+sin(ay) [C(ay) - S(ay)]}
(a + x)-l (2n)! y-l {[ ~ - C (a y)] cos (a y) +
+ [~ - S (a y)] sin (a y) }
cos (ay) Ci(ay) + sin (ay) [~ n + Si(a y)]
(a - xtl Das Integral ist als CAUCHY-Hauptwert definierl.
0 O<xb xC(ay+ by) - 2 sin (ay) S(ay+ by)]
(a + x)-I 2a-l - (2ny)I{[1-2S'(Vay)] cos (ay) -
- [1 - 2C (Vay)] sin (ay)}
0 O<x<a n!(2y)-![cos(ay) - sin (ay)]
(x - a)-l x>a --
X-I (x - a)~ x>a -nal [1 - C(ay) - S(ay)]
§ 1. Algebraische Funktionen 3
0 O<x<a na-![1-C(ay) - S(ay)]
X-I (X - a)-l- x>a
0 O<x<a n(2a)-1 {[1-C(2ay)-S (2ay)] cos (ay) +
(x-a)-l(x+aJ-I x>a + [C(2ay) - S(2ay)] sin (ay)}
(a - x)-~ O<x<a (2yny [cos (ay) C(ay) + sin (ay) S(ay)] 0 x>a
0 O<xb - S(ay+by)] cos (ay) + [C(ay+by)-
- S (ay + by)] sin (a y)}
(a 2 + X2)-1 ~ na-1e-ay 2
x(a2 + X2J-l - ~ [e-aYEi(ay) + eaYEi(- ay)] 2
b [b2 + (a - X)2]-1 + ne-bYcos(ay)
+ b [b2 + (a + X)2]-1
(a + x) [b2 + (a + X)2]-1 + + (a - x) X ne-bYsin(ay) X [b2 + (a - X)2J-I
~na-lsin(ay) 2
(a2 - X2J-l Das Integral ist als CAUCHy-Hauptwert definierl.
cos (a y) Ci (a y) + sin (a y) Si (a y)
x (a2 - X2J-l Das Integral ist als CAUCHy-Hauptwerl definiert.
x-i (a2 - X2J-l ~ na-~sin(ay)+(~ nyY'a-1SO,!(a y)
Das Integral ist als CAUCHY-Hauptwert definiert.
(a2 + x2)-! Ko(ay) 1*
4 I. FOURIER-Kosin us-Transformationen
[X + (a2 + X2)~J-l a2 y-2_ (ay)-lKl(ay)
x-!(a2+ x2)-l (~ nyt' L!(~ aY)KiC a y)
(a2 _ x2)-l O<x<a 1 -nlo(ay)
0 X> a 2
x(a2- x2)-! O<x<a a[1- ~ nH1 (a y )]
0 X> a
(x2 - a2)-! - -n Yo (a y)
X> a 2
0 O<x<a -n a 1 1 - -nay [lo(ay) H_l (ay) + 1 _ { 1 2 2
X-1 (X2- a2)-l x>a + Ho(ay) .h(a y )]}
x-1.(a2 - X2)-~ O<x<a C nytyl[J-!(~ ay)r 0 x>a
0 O<x<a - (~ n)l yl J-l( ~ a y) Y-1( ~ ay) x-i(x2- a2)-i x>a
(a2+ x2)-1[(a2+ X2)!+ aJ! n!(2y)-!e-ay
(a2 +x2)-! [(a2 +x2)! +a]-l n(2a)-! Erfc (Vay)
x-i(a2 + x2)-l X 2-ina-le-lay JoC a y)
X [x+(a2+x2)l.J-!
x-~ (a2 + X2) -~ X r i a-2sinh (~ a y) KIC a y)
X [x + (a2+ x2)lJ-ft
x-lea + X + (2ax)l]-1 n(2a)-l eaYErfc (Vay)
[(a2+ x2) (b2+ X2)J-l ~n(a2 - b2)-1 [b-1 e-by - a-I e-aYJ 2
(a4 + x4)-1 ~ na-Se-2-iaYsin(: +2-l a y)
§ 2. Beliebige Potenzen
- ~n< (j<~n 2 2
x2 [x4+2a2 x2 cos (2 (j) +a4]-1
-~n<(j<~n 2 2
xli (A 2+A1 )11 AIA2 A 2-A1
Al = [a2+ (b - X)2]!
X2m(a2n +X2n)-1 n> m2;O
n ~ nn-la2m-2n+l L e-aysin [(2k-l) "/(2n)] X
k=1
X cos [(k- ~)n-ln]}
Rev>O
Rev> -1
Rev> -1
o
sine nv)r(1 - v) y.-1
~ a' V-I [1 Fl (v; V + 1 ; i a y) + + 1 Fl (v ; V + 1 ; - i a y) ]
~ i y-.-l [ei(a YH "') y (v + 1, i ay) -
- e-i(ayH"·)y(v + 1, - iay)]
5
Re(v,p) > 0
0 x>a
-~<Rev<~ 2 2
0 x>a
O<Rev<~ 2
+ 11\ (v; v + p; - iay)J
()V( )f{ r(t+tv) S () 2a a:n;y r(-tv) -v-l.1 ay -
r(1+tv) (} - 2 r(-t-tv) S-v-a.l ay)
~:n; av- 1 sec(~:n;v ) cosh (a y)+:n;l2V- 2y1-v X
xr(~ v- ~)[r(1- ~ v)r x
F. (1 . 1 1. 3 1. 1 2 2) Xl 2 --v ---v -a y , 2' 2 2' 4
:n;1( ~ y/an r( ~ + v)rKv(a y)
2v-1:n;l r( ~ +v) aVy-V ]" (a y)
- ~:n;l r(~ -v) (~LrY(ay) 2 2 2 a v
v+1 -v S ( y) - a Y v-1.v+1 a
= ~ (v + ~ f1 a2P+1_ 2v-1:n;! av+I X
X r(~ +v)y-vHp+I(ay)
r v- 1:n;! ai-v r (~ - v) y' ],,-1 (a y)
§ 2. Beliebige Potenzen
I(x) g(y) =JI(x)cos(xy)dx o
o I ~ n sec (nv) a-h - 1 X O<x<a 2
X-1(x2_a2)-v-! x>a
-1 <Rev <~ 2
-~<Rev<~ 2 2
o x> 2a
-H.(ay) ],,-1 (a y)]}
- : nlr(: -v) (2a)-V yVX
X []" (a y) sin (ay) + Y" (a y) cos (ay)]
- : n! (;a r r C - v) X
X [Y" (a y) cos (a y) - ]" (a y) sin (a y)]
nlrr(: +v)(;arV cos(ay)J.(ay)
+: i[Jv(iay)-Jv(-ia y)]}
Rev> -1
n esc (nv) a-v [~Jv(iay) +~ J v( -iay)- 2 2
x-i (a 2 + x2) -! X
X [(a2 + X2)!+ x]"
Rev < 1., 2
Rev> _1., 2
- cos (: nV)Iv(a y)]
(: nyY aV L!(!-v) (: ay) K HHv) (: a y)
8 1. FOURIER-Kosinus-Transformationen
-(a + ix)-V]
1 1
- 2 2
X W~v,! (a y) M_lv, -1 (a y)
a-I r(: - ~ v) (2y)-l X
X Wi>, 1 (a) M_lv, -1 (a y)
n (- 1)n (2n)! [r(v)]-I X
X e-ay y.-2n-I L~-;;2n-I (ay)
in(-1)n(2n -i)! [r(v)]-I X
1---------------
+ [(a2+ X2)!- x]"}
-1<Rev<1
-1 < Rev < 1
+ [x - i(a2- x2)ly}
O<x<a
naV [sin (a y - ~ nv) J.(ay)
- cos (ay - ~ nv) Y" (a y) 1
~naVsec(~nv) [J.(ay) +J v(ay)] 2 2 -
§ 2. Beliebige Potenzen 9
+ [a - (a2 - X2)~y}
-~<Rev<~ 2 2
Re(fI"v) > -1
a<x
x>a
g(y) = il(x) cos (x y) dx o
2a 2 -+-'V ---v X ( )-' B(' 1 1 1) 4 2' 4 2
R(' 1 1 .) Xl 1 4-T V; T; -zay X
R(' 1 1· ) Xl 1 4-T V; T; zay
~ av+2.u+ l B ~ + 1, ~ v + ~) X
x lF 2 -+-v'- fI, +-v+-'--C 1 1 1 3 _a2y2)
2 2' 2' 2 2' 4
-na'[sin(~ vn)J.(ay) +
+ cos(~ vn)y'(a y)]
X [J-Hi-V) (~ ay) Y-!(Hv) (~ ay) +
+J-HHV)(~ aY)Y-Hi-")(~ a y)]
1. [( 1 )r TnaV-lsm (a y) +nl 2vav- l r -Tv X
xr(~ + ~ v) yl S_!_v,!(ay)
(-1)nn!(2a)-V[r( ~ + v)r :;:n X
X [yV K.(ay)]
10 I. FOURIER-Kosinus-Transformationen
xV(a2+ X2)-1'-1
o
1 0-21'-1 B (1 + 1 1 1 + ) 2 a 2 2'1', 2-2'1' P, X
x 1R2 -+-'1' -+-V- Il - -- + ( 1 1 "1 1 1 " a2 y2 )
2 2' 2 2 r' 2' 4
+nt20-21'-2[r(1+p,-~ V)r X
Xyl+21'-Or(~ '1'- ~ -p,)X
_ ~ V" a2y2 ) 2 ' 4
Rev> -1
~ nl(a2+ y2)-icos[~ arctan(~)l
(~ ny (a2 + y2)-i [a + (a2 + y2)lJ!-
n! an+1(a2+y2)-n-l L (_1)m(n+1) (~rm O:;;2m:;;n+l 2m a
(-1)"(~ny ~X 2 dan
x{(a2+ y2)-I[(a2+ y2)1_ a]-I}
r(v) (a2 + y2)-lv cos [V arctan (~) 1
r( 1 + v) (a2 + y2) -i(1+v) e-ab X
X cos [by+ (v + 1) arctan(~)]
§ 3. Exponentialfunktionen 11
e-azl
x1e-ax'
X-ie-ax-
x-2"e-a-xl
g(y) = jf(x) cos(xy)dx o
r(v){y-vCOSC ~V)+ ~ (2a)-VX
X [, (V 1 + i Y ) +, (V 1 i Y ) '2 - '2- - -2a 2a
- ,(V, i :a) -, (v, - i :a )]}
~ y-2 - ~ (: r [CSCh (~ ~) r logY-~[tp(iy) +tp(-iy)]
2
a log (~2::~2) -y arccotan[~ (~)3 + %(~)l
~ b-1[B(v, a~iY) + B (v, a~iY )]
y' 1 -4; -~ia-le 2
1 (Yy_L[ (y2) (y2)] 4~ 2a e Sa Ll g; - 11 g;
1 (yy-:: (y2) -~ - e L 1 -2 2a Sa
y' (-1)"~!2-"-1a-2"-1e-4a'He2n(2-i ~)
/(x) g(y) = jl(x) cos(xy) dx o
xVe-ax' 1 -!-!v r( 1 + 1 ) T a T TV x Rev> -1 e 1 1 y2) X1F1-+-v'-' --2 2 I 2 I 4a
(b2+ x2t 1e-a' x' : n b-1 ea'b' [e- bY Erfc (ab - :a) +
+ eby Erfc (ab +:J] -
+ e1b-1 (a+i y)' Erfc [~ b-i (a + i y)]}
1 ~ r(v) (2b)-lv e8b(a'-y')
xv-1 e-a x-b x' x{e-i~~ D_v[(2b)-l(a - iy)] + Rev>O
+ ei:~ D_v[(2b)-l(a + iY)J}
a (:Ye-(2a y)1 COs (2ay)1 x-~e-x
a (:t {/"--i- Kv[2(iay)!] + x-v-1e- x
Rev> -1 . v + e-"'4 K v[2(-iay)!]}
nl(a2+y2)-!e-2bu X
U = r![(a2+ y2)1+ a]l
v = r! [(a2+ y2)1_ a]!
n! b-1 e-2bucos (2b v) b'
x-i e-ax- x U = 2-l [(a2+ y2)l+ a]!
v = rl (a2 + y2)1_ a]!
§ 3. Exponentialfunktionen 13
b' b'{(a + iy)-!' K.[2b(a + iy)!J + x·-Ie-ax-X-
+ (a - iy)-!' K.[2b(a - iy)!]}
00
x-2e-a' r ' 1 a-I L (_ay)n -n 2 n=on!r(t+tn)
e-ax-1 ( ~ n Y a y-~ {cos (4a; ) [~ - C( :; )] +
+ sin (:;) [~ - S ( :; )]}
- sin (:: )[~ - C ( :: )]}
1 2"n Ii II8y sin 8y+s -
x-I e-ax" - Yl ( :: ) cos (:: + ;)]
XD_2'[~ ay-i(1-i)]+ Rev> 0
e-b (a·+x.)/t ab(b2 + y2)-ilKI[a(b2+ y2)liJ
(a2 + x2) -I! e-b (a'+x')! K o [a(b2+ y2)!J
1 en YY L 1g a [(b2+ y2)!_ bJ}X
x-l(a2+ x2)-le-b(a'+x')" XK!{~ a[(b2+ y2)l+ bJ}
1 (2bnY K!{~ a[(b2+y2)!_YJ}x (a2 + x2) -i e-b (a'+x')"
X Kl ~ a[(b2+y2)!+yJ { }
(a2+ X 2)-!X
X [(a2+ x2)i+ a]-! X n(2a)-!eabErfc{ai[(b2+ y2)!+ b]!}
X e-b (a'+x')!
X [(a2 + X2)!- a]-Ix X (a2 + X2) -! e-b (a·+x·)l
(a2 + X 2)-! X
X e-b(a'+x.)t
X e-b (a'+x')!
x-2.-1(a2_ x2)-!x
X eb (a'-x')! +
X e-b (a'-xI)!}
Re(: ±v»O
r(: + ~ v)a-1 (2Y)-!X
X W_!v,! {a [(b2 + y2)t + bJ}
- : (nb)! [li(Zl) Y-!(Z2) + 1!(Zl) I-!(z2)]
= - : (nbl ) [J-! (Zl) y!(Z2) + + Y -1 (Zl) Ii (Z2)]
Zl =~ a [y + (y2 - b2)!] 2
Z2 = ~a [y - (y2 - b2)!] 2
y> b
x~,_!{a[b + b2- y2)!J} X
xM_., -1 {a [b - (b2 - y2)!J} b> Y
RAMANuJAN, S.: Mess. Math., Bd. 44, S.75-85. 1915.
§ 4. Logarithmische Funktionen 15
x-!log X
X-I log (1 + x)
log(a-x) O<x<b
..!..na-1[sin(ay) Ci(ay) - cos (ay) si(ay)] 2
~ n a-I {sin (ay) [Ci(ay) -log(ab)]-
- cos (ay) si(ay)}
- e-aYEi(ay)]
~ {[ Ci ( ~ y) r -[si ( ~ y) n ~ {[ci( ~ y)r + lsi( ~ Y)r}
y-lg n[cos(by) - cos (ay)] +
+ cos (by) Si(by) + cos (ay) Si(ay)-
- sin (ay) Ci(ay) - sin (by) Ci(bY)}
y-l{sin(by) log (a - b) + sin (ay) X
X [Ci(ay) - Ci(ay - by)] -
- cos (ay) Si(ay) }
.x·-1log x T(l') y-' COS (~ 7& V) X
0< Rev < 1 X [11' (v) - ~ 7&tanC 7&v)-logy]
(a + i x)-llog(a + i x) + 7&e-ay (y + log y)
+ (a-i x)-llog (a-i x)
+ yarctan(~)]
x·-1e-axlog x {cos [v arctan ( ~ )][11' (v) - ~ log (a2 + y2) ] -
- sin [v arctan ( ~)] arctan (~)} X Rev> 0
xT(v) (a2+ y2)-1'
e-ax (log X)2 x[ay+ ~alog(a2+y2)] +
+ 2yarctan (~) - a [arc tan (~)n
X-I e-1a' r 'log x i: [Ko (a V- iy) -Ko(aViY)]-
-log(~ VY)[Ko (a Viy )+Ko(aV -iy)]
log(a2- X2) O<x<a y-1sin(ay) [Ci(2a y) -y -log( ~ y)] -
0 x>a - y-1cos(ay) Si(2ay)
( a2 +x2) log b2+X2 7&y-l(e-by _ e-ay)
I a2+ x2 1 log b2 _X2 7& y-l [cos (b y) - e-a YJ
X-I log (1 + ::) Ei(- ay) Ei(ay)
§ 4. Logarithmische Funktionen 17
(a2 + X2) -!log (a2 + X2)
0 O<x<a (X2- a2)-~log(x2- a2)
x>a
O<x<1 0 x>1
log(1 + a X-I)
0 O<x<a
log [ H (1 ~aX-l)!]
x>1
(a2 - X2) -k log (Va2 - x2 l O<x<a
0 x>a
n y-l (1 - e-ay)
ncosy [+n Yo(y) - (y + log 8y) Jo(Y)]
y-l [~n + cos (a y) si (ay) - sin (a y) Ci (a y)]
ny-I [1- cos (ay)]
~ny-l[1 + Lo (a y) - Io(ay)] 2
: n y-l [1 - cos (~ a y) Jo (~ a y) -
-sin(~ aY)Yo(-i-ay)]
~2Yo(ay)_: Jo(a y)[y+log(2;)]
n2 - [Io(ay) - Lo(ay)] 4
(a2+ x2)-lx .!.. [5_ 1 o(iay) + 5_1 0(- iay)] + 2' ,
X log [x +(a2+ x2)1] + log a Ko (a y)
.!..a y-2 -.!..:n; y-l csch (~) 2 2 a
§ 5. Trigonometrische Funktionen
1 y<a -:n;
0 y>a
xv- 1 sin (a x) n [ C) r 4 cos T:n;v r(i - v) X
-i<Rev<1 x{(y + a)-V_ sgn(y - a) Iy - ai-v} .--
~:n; e-ab cosh (b y) y<a x (x2 + b2)-l sin (a x)
2
- ~:n;e-bYsinh(ab) y>a 2
~:n;b-2[1 - e-abcosh(by)] y<a X-I (x2 + b2)-1 sin (a x)
2
.!..:n; b-2 e-b Y sinh (a b) y>a 2
1 1
e-bx sin (a x) -(a + y) [b2+ (a + y2)]-I+ - (a - y) X 2 2
X [b2 + (a - y)2]-1
x-1e-x sin x .!.. arctan (2 y-2) 2
X-I sin2 (a x) .!..logI1- 4a2y-21 4
X-I sin (a x) sin (b x) 1 (a+b)2_ y2 -log
(a_b)2_ y2 2
-i-{(y + 3a) log(y + 3a) + (y - 3a) X
xlogly - 3 al- (y + a) log(y + a)- - (y - a) logly - al}
~(3a2- y2) 8
O<y<a
° y> 3a
(_1)mr2mmn{(m!)-2y2m-l +
+ i: (_1 t [(2an+ y)2m-l+(12an_YI)2m-l]} n=l (m+n) ! (m-n)!
y::S;2am
F(a) = L (-1tX n=O
X [(2n+1)a+y]2m+ [(2n+1) a_y]2m
(m+ 1 +n) ! (m-n) !
F(a) = L (-1tX n=O
[y+(2n+1) a]2m_ [y-(2n+1) a]2m X (m+1+n)! (m-n)! +
m +L(-1)n X
X [(2n+1) a+y]2m+ [(2n+1) a_ yJ2m
(m+ 1 +n)! (m-n) !
20 1. FOURIER-Kosinus-TraIlsformationen
n = 0, 1, 2, ...
Rev>O
(a2+ x2)-llog(c2sin2bx)
x·-1 cos (a x) 0< Rev < 1
(b2 + X2)-1 cos (a x)
e-b x cos (a x)
e-bx' cos (a x)
21-·COSC y)r(v) [re + ~ '1'+ 2:)X X r (~+ ~ '1'_ L)]-l
2 2 2n
m=1,2, ... , m:;;;' :b <m+1 In
Fiir m = 0, LO =0 n=l
~ log I 1 - a2y-2\ 2
1 -:n;(a - y) 2 y<a
0 y>a
~:n; b-1 e-ab cosh (b y) 2 y<a
~:n; b-1e-b y cosh (a b) 2 y>a
~ bUb2 + (a - y)2]-1+ [b2+ (a + y)2]-1}
1 (nt - a'+Y' (a y ) 2 b e 4b cosh 2b
§ 5. Trigonometrische Funktionen 21
Ibl < 1
Ibl < 1
Ibl< 1
Ibl < 1
0 x>1 Rev> 0
(cosh a-cos X)-1 O<x<n 0 x>n
(cosh a-cos x)-' O<x<n 0 x> n
(cos x-cos (5j"-! O<X<!5 0 X>!5
0<15 < n, Rev> -~ 2
x-21og (cos2 a x)
O:;;;;y<1
X (be-aY_bn+1e-Aa) +
y=n+it, O:;;;;it< 1, n =1, 2,), ...
~na-1(ea- b)-1cosh(ay) 2
+ (e-a_ b)-1(e-an- aA _ bne-aA)]
21-. r(v) [r(~ + ~ v+ ~)x
C 1 y)t Xr 2+2v--;t
_ ys~n(n-.& ~ (-1)ne (n2_y2)-1e-na sInha n
n=O
- 21-. [r(lI)J-l e(.-l) a (sinh all-a. X
X . ( ) f (_t)n Ene-na r(n+v) X ysm ny 2 2 . I n=O n-y n.
x 2F1(1-v, n+1-lI; n+1; e-2a)
(~ ny r( ~ + v) (sin 15)' P-=-{+y(cos!5)
n[Ylog 2 - a +n~1 (-1tn-1(y - 2a n) 1
m:s;;L<m+1, 2a
m = 0, 1, 2, ...
g(y) = j,(x) cos(xy) dx o
:na-1 {cosh(a y) log (1 + e-2ab) + m
+log(~ c)e-aY+n~I(-1)nn-IX
X sinh [a(y - 2bn)J} m
m = 0, 1, 2, . . . Fur m = 0, 2: () = 0 n=1
:na-3 {(a y + e-ay) log 2 - cosh (ay) X m
X log (1 +e-2ab) - ab _. 2: (-1)n n-1 X n=1
X [sinh(ay - 2abn) + sinh(2abn) -
:na-l{[~!5+ ~ log(~ c)]e-ay +
+ cosh (a y) log (1 ± e-"-ab) +
xi: (+ 1)nn··1 e-n " sinh [a(y - bn)J} n=1
m
m = 0, 1, 2, . . . Fur m = 0, L () = 0 n=1
:n cosh (by) (1 + e2abt 1
:ncosh(by) (e2ab _1tl
'" f(x) g(y) = Jf(x)cos(xy)dx 0
~:n;cosh(ay) csch(ab) 2
sin (a x2) : (2:t[cos(::) -sin (::)]
sin [b (a2 - X2)] O<x<a ~ (:Y U.(2a2b, ay) 0 x>a
sin[a(1-x2)] 1(ny ( n y2) -- - cos a+-+-2 a 4 4a
X-2 sin (a x2) ~ :n;y[S(::)-C(::)] +
+ (:n; a)! sin (~+~) 4 4a
X-I sin (ax2) ~ :n;g -[C(::W-[S(::)n
n(y)~[ (n y2) (y2) - - cos -+- J_I -- xl sin (ax2)
4 2a 8 Sa 4 Sa
-sin(; + ::)J1(::)] x-lsin (a X2) - ~(-Ltsin(~-~)J-,(~) 2 2a 8a S ~ Sa
xp-l sin (ax2) 1 pC) . [-iTP ~ (1 1 . y2) 4 a-! r 2 v Z e IFI 2v;2;~4a-
-2<Rev<2 i~p C 1 . y2)] -e 4 11\ -v'-' -.-~- 2 '2' 4a
~ :n;!(a2 + b2)-le-laY'(a'+b')-' X e-ax' sin (b x2) 2
, [1 (b) by2 ] X sm 2arctan a - 4(a2+b2)
o
cos [b (a2 - X2)] O<x<a ~ (~yU!(2a2b, ay) 0 x>a
cos r a (1 - X2)] - - sm a+-+-1(ny,( n y2) 2 a 4 4a
n ( y )~ [ ( n y2 ) ( y2 ) - - cos -+- I, - + x!, cos (ax2) 4 2a 8 8a ., 8a
+ sin (; + ::) I-I (::)]
x-l cos (a X2) n(yy (y2 n) (y2) 2- 2a cos ~-8 1-1 ~
x·-1cos(ax2) 1 -1 C)[ -;~. c 1, yZ) 4 a 'r 2'1' e 4 11\ 2V;2;~4a +
o <Rev <2 ;~. (1 1 ,y2)] +e 4 11\-V'-'-~-2 '2' 4a
-.!.. nl (a2+ b2)-i e-!aY'(a'+b')-l X e-ax' cos (bx2) 2
X cos [~by2(a2 + b2tl-~arc tan(:)]
~ (3a)-~yl{3IiKl[2(3a)-~y~J + cos (a3 x3) + nI~ [2 (3 a)-~ y~J +
+ nI-l[(2(3 a)-i yIJ}
sin (:) - ~ (~rl[nYl(2(aY) + 2K1(2Vay)]
x-1 sin (:) ~ n]o(2Vay)
+ cos (2Vay) - e-2Yay j
_8 ' (a) ~ (2:Ylsin (2Vay) + X 'sm x + COS (2VaY) + e-2]IaY]
xv-l sin (;)
-Lv(2Vay)]
-----1'------~--~
X-I log (bx) sin(;) -i-n [Jo(2(aY)log(b Vf) - Ko (2Va y)]
X-I cos (; )
- sin (2Va-y) +e-2 Va y]
1 (1 )(Y)-k V 4 ncsc 2 vn ~- X
X [J-v(2Vay)-fv(2Vay) + xv-lCOS (:) + Lv(2 Vay) - Iv (2 Vay)]
_ 1 < Re v < 1 I = -} n ( ~ t1 v {cos (: v n) X I I X [~ Kv(2Vay) - y'(2Vay)]-
-------1 -sin(: vn)fv(2Va y)}
'" f(x) g(y) =ft(x)cos(xy)dx 0
sin [b (a - x)l] O<x<a ~ ni by-! Di(2ay, b all 0 x>a
x-isin (a xi) (2;y[ C (::) sin (::) - s (::) cos (::)]
-- 1 (a) ~ [ ( a2 n) ( a2 ) --n - sin --- 1-1 - +
x-1sin(axi) 2 2y 8y 8 8y
+ cos (:: - ~) 1£ ( :: ) ]
x-!sin(axi ) ( a t ( a2 3n) ( a2 ) n - cos --- I i - 2y 8y 4 8y
~ n! a (b 2 + y2)! e-1a' b (b'+y')-l X e-bx sin (a x!) 2
[ a2y -; arctan(~)] X cos 4 (b2+y2)
X-I e-ax~ sin (a xi) ~nErf[a(2Y)-§] 2
x·-1 e-axi sin(axi _ ~ 'lin) - (~ n y (2 y) -. e-ia'y--l Dh - 1 (a y-i)
Rev> 0
(a - x)-!cos [b(a - x)~J
(;YU!(2a y, bal) O<x<a 0 x> a
x-i cos (a xl) (ny . (n a2 ) y SIll 4+4Y
x-1cos(ax!) 1 (a l [. (as n) J ( a2
) - 2 n 2Y SIll BY + 8 -1 BY +
+ cos (:: + ~) 1-1 (:: ) ]
x-I cos (a xl) ( a y ( a2 n) ( a2 ) n - cos - - - 1-1 -
2y 8y 8 8y
x-v sin (a xl)
0< x b
(x2 + C2)-1 (a2 + x2) -i X
X sin [b (a2 + X2)!]
g(y) = f!(x) cos(xy)dx o
n~ (a2+ y2)-! e-!ab'(a'+b')-l X
[ b2 y X cos (2 2) 4 a +y - ~ arc tan ( ~ ) 1
- ~ ar (~ - V) yV-~ X
x[e-i-'i(V+~) R (~-v·~·-i~)+ 1 1 2 ' 2' 4y
+ if (v+t) R e . 3 .. a2 ) 1 e 11--V-1- 2 ' 2' 4y
~ yv-1 r(1 - V) i X 2
[ -i v-"- ( 1 . a2 ) xe 2 11\1-V;2;-1 4Y -
i v-"- ( 1. a2 ) 1 -e 21F1 1-V;2;14Y
ani (2 y) -I e-~a'y-l
( n Y -'a'y-' - e 11 2y
ncos(~ bY)Jo[~ b(a2 + y2)!.]
(yr' - n "~(2Y, a)
O<yb
2n b c-1 e-CY c=a
y:2;,b
X sin [b (a2 + x2)1]
x(a2+ x2)-l X
X sin [b (a2 + x2)l]
{[(a2 + x2)1 + x)" + + [(a2+x2)i_x]"} X
X (a2 + x2)-1 X
-1<Rev<1
o
(~ n)iylI_lg a[b-(b2- y2)!]}x
X It{ ~ a [b + (b2 - y2)l] } y b
= _~na.[(b+Y)l' + (b-Y)lV] X 2 b-y b+y X {cos (~ vn).T.[a(b2-y2)t]-
- sin (~ vn) ~[a (b2 _ y2)!]}
O<y b
- ~n Yo [a(b2- y2)l] 2
Ko [a (y2 - b2)l]
XYlg a[b+(b2- y2)!]} O<y<b
{[(a2+ X2)! + x]" + + [(a2+x2)t_X]V} X
X (a2+ X2)-~ X
-1 <Rev <1
(a2 -x2)-lsin [b(a2-x2)!J
O<x<a
O<x<a 1
_ x- ~ (x2_a2)-l! e-b (x'-a')"
(~)t (y2 _ b2) -! e-a (Y'-b')~ X 2y,
( 1 . b) X cos 2 arc SIll -y-
= _ ~ nav [(~)!V + (~)!Vl X 2 b-y b+y
X {sin (~ v n) Iv [a (b2 - y2)! J +
y>b
= aV cos (~ v n) [( ~ ~: t + ( ~ ~: t 1 X
y>b
(~ n)~b~Id~ a[(b2+y2)!;_yJ}x
X ld ~ a [(b2+ y2)! + yJ}
(~ ntY!;I-t{~ a[(b2+y2)l!- bJ}X
X Y_!g a[(b2+y2)?!+ bJ}
30 I. FOURJER-Kosinus-Transformationen
ex> I(x) g(y) = fl(x) cos (xy)dx 0
(a2 - X2) -l sin [b (a2 - x2)IJ
O<x<a ~nYo[a(b2+y2)!] (2 2) -! -b (x'-a')~ 2 - X -a e
x>a
(a2_ x2)-lcos [b(a2- X2)!] ~ n Jo [a (b2+ y2)l] O<x<a 2
0 x>a
(a2-x2)-icos [b(a2-x2)l] (~ n)ibiJ_!g a[(b2+y2)i_y]}x O<x<a
X J-1g a[(b2+ y2)l + y]} 0 x> a
x-! (a2 - x2) -l X (~ n)iy!iJ_d~ a[(b2+y2)~-b]}x
X cos [b (a2 - X2)!]
O<x<a xJ-1g a[(b2+y2)1+b]} 0 x> a
0 O<x<a x (b2 + x2 - a2)-1 X ~ n e-bc cos [y (a2 - b2)!]
2 O<y<c
X sin [c (X2- a2)!] x>a
0 O<x<a - (~ n y bi It g a [y - (y2 - b2)l] } X (x2 - a2)-: sin [b (x2 - a2)!]
x> a X y! g a [y + (y2 - b2)!i] } y>b
(a2 _ X2) -i e- b (a'-x')!
O<x<a 0 O<yb
X sin [b (X2 - a2)l] 2
x>a
0 O<x<a Ko [a (b2 _ y2)!] O<yb x>a 2
X COS [b(x2 - a2)iJ
0 x> n
(cos xl-it e-2acosx
o <x <!!.. 2
0 x>~ 2
O<x < .'!. 2
0 x>~ 2
(cos x)-! e-asecx
O<x<!':. 2
0 x> !':. 2
0 x> n
X y!{~ a[y+(y2-b2)lJ} y>b
+(nYyLd~ a[b-(b2- y2)lJ}X
X K!g a[b+(b2- y2)!J} y <b
n(na)lcos(~ ny)x
- I! (!_y) (a) It(Hy) (a)J
n(na)l! cos (~ n y) X
X [I -HHY) (a) L,W-y) (a) +
+ I~ (~_y) (a) I~ (Hy) (a) J
~ n(na)! [Li(Hy) (a) Li;,(!_y) (a)-
- I!dHY) (a) I! (!_y) (a)J
~ n (n a)! [I -l (Hy) (a) I _~ (}_y) (a) +
+ I!dHY) (a) Il;,(!_y) (a)J
(1 + cosny) SO,y (a)
cos (a sin x) O<X<7C 0 x> 7C
sin (a cos x) O<x<~ 2
0 x>~ 2
0 x>~ 2
O<x<~ 2
0 x>~ 2
O<x<~ 2
0 x> !!.- 2
O<X<7C 0 x> 7C
OJ g{y) = If (x) cos {x y) d x
o
cos(~ 7C Y)SO,y(a) . = : 7CCSCC 7C Y) [Jy(a) -J_~.(a)]
-ysinC 7C Y)L1,y(a)
~ 7C (7Ca)! I-!{l-y) (a) I-HHy) (a)
7C (a 7C)~ cos (~ 7C y) Ii (k-y) (a) I! (Hy) (a)
(sin x)-!cos(2asin x) I
0< x < 7C I 7C (a 7C)t sin (~ 7C Y) I-Hl-y) (a) I-HHy) (a) o X>7C
I{x)
arcsin x
g{y) = f/{x)cos{xy)dx o
~ 7C y-l Ho (y) 2
§ 6. Zyklometrische Funktionen 33
o <x <1
o x>1
X cos (v arc cos x)
o <x<1
0 x>1
Rev> 0
-1 <Rev <0
-1 < Rev <0
O<x<a (X2 - a2)-! X
X arc tan [a-1 (x2 - a2)~]
x> a Oberhettinger, Tabellen
~ n (~ nyY Jk(v-~) (~ y) J-~(vH) (~ y)
- ~nEi(-ay) 2
X sinh C y) - I vH (~ y) cosh (~ y)]
- n-!; F(v + 1) y-v-~ sin (v n) X
X cosh ( {- y) KvH (~ y)
~ y-l [e- ay Ei(ay) - eay Ei(- ay)] 2
ny-le-(4a)~y sin[(~ a/ y]
3
g(y) = jl(x) cos(xy) dx o
1 -1~( 1)m -aysin[(m-t).:!.] -2 ny L.... - e " X m=1
X sin [ a y cos (m - ~) :]
~ ny-1{cos(a2y) [C(a2y) - S(a2y)]
§ 7. Hyperbolische Funktionen
n=O,1,2, ...
-.!... n a-2 y csch (~) 2 2a
-.!... na-3 (a2+ y2) sech (~) 4 2a
22n -
1na-2 y csch (~) IT [(L r + m2] (2n-1)! 2a m=1 2a
2 2n
X r(-.!...v - i--"-) 2 2a
2'na-1sin(~ nv)r(1 - v) X
X[r(1- ~v+i ;Jx X r (1 - ~ v - i :a) r1 cosh ( ~~ ) X
X [cosh ( n:) - cos (nv) r1
§ 7. Hyperbolische Funktionen 35
[cosh (ax) + cos b]-1 na-1 sinh (b:)
b<n sin b sinh ( n: )
[cosh (ax) + cos b]-i Z-! na-1sech (nL)P_..!c+iL (cos b) a 2 a
[cosh (ax) + cosh b]-1 na-1 sin (b:)
sinh b sinh ( n: )
[cosh (ax) + cosh b]-l Z-!na-1sech (~) P _..!c+i L (cosh b) a 2 a
-1 COSh(n:)sin(b:) -na
(cosh x - cos <5)-! Z-k [Q -Hiy (cos <5) + Q -k-iy (cos <5)]
O<<5<n
(a + cosh x)l- (b+ cosh x)l i 2-i n y-l sech (ny) [~+iY (b) - ~-iY (b) -
a, b> -1 - I!+iy (a) + I!-iy (a)]
[1 +2 cosh (x lPf)rl (~ ny[1 +2 cosh (y W)r1
(cosh a + cosh x)-'" (~ nt [rGu)]-1 (sinh a)!-'" r(f-l + iy) X
Ref-l> 0 X r(f-l - i y) \l3~-f+iY (cosh a)
(cosh x + cos <5)-'" (~ nt [r(f-l)]-l (sin <5)l-,.. r(f-l + iy) X O<<5<n
Ref-l> 0 xr(f-l-i y) P!"i+iY (cos <5)
(cosh a - cosh x)-'"
O<x<a (~ ny r(1- f-l) (sinh a)l-'" \l3~!tiY (cosh a) 0 x>a
Ref-l <1 3*
0 O<x<a (cosh x - cosh a)-JJ-l (2n)-lr(~ -,u)(sinha)-JJe-iJJ:1tx
x> a • X [O':l-iy (cosh a) + O':Hiy (cosh a)]
1 1 -- <Re,u <-
n b-I cos (~:) cosh (~~) cosh (bx)
cos (nba) + cosh (n:)
. (na) sinh (ax) ~nb-I
: b-I{2nsin (~a) X
sinh (ax) O<a<b
X [cos (nba) + cosh (nnri + cosh (bx) + "P eb-:b+iY ) + "P e b-4ab-iY )-
__ e b+a- iY ) _ eb+a+iY )} "P 4b "P 4b
cosh C ax) ~na-I cos (b:)
2 COSh(~ b)COSh(n;) cosh b+cosh (ax)
cosh (~ ax) ~na-I
cos !5+cosh (a7) 2 cos(~ b)COSh(;) O<I5<n
xcsch(ax) Rea>O : n 2 a-2 [sech( ~:)r
X-I (csch X - X-I) -log(1 + e-"Y)
X-I (1 - sech X) log [r(~+itY)ni--itY)] -log (~Y) r(t+i t y) r(t-i -~ y) 4
§ 7. Hyperbolische Funktionen 37
-1 <Rev <2
[sinh (ax)]2 x sinh (bx)
cosh(ax)-cosh(bx) x sinh (ex)
~1J!(~ +i~y)+ ~1J!(~ -i~Y) -log (~ y)
rv rev) X
~lo [COSh(*)+Sin(*)]
~ 10 [cos (~)+ cosh (-¥-) ] 2 g cos(:n:ea)+COSh(:n::)
2a3 [sech(ay)J3
- e-Y arctan (elY)
~ ye-Y - ~ + log (1 + e-Y) cosh y 2 2
1 00
2 na-lsec (a b) e-ay - b n-l L: (-1)m X m=O
[( 1)2 (ab)2]-1 -(m+-})": X m+- - - e 2 :n:
!(x) g(Y)~rt(x)cos(xy)dx o ,
x-Ie -a x sinh (bx) ~ 10 [y2+ (a+b)2] 4 g y2+(a_b)2
1 1[ (iY) ( iY) 1-tanh(ax)
4 a- 1p ~ + 1p - ~ -
X-I tanh (a x) log ctnh ( ::)
x (1 + x2tl tanh (~ n x) - y e-Y - coshylog(1 - e-2y)
x(1 + x2t1tanh (17,;) cosh y log ctnh (;) - ~ ne-Y -
- 2 sinh y arctan (e-Y)
~ a n~ (b2 + y2)i e!a'b (b'+y')-' X e-bx sinh (ax!) 2
X cos[~ arctan(~)-: a2y (b2+y2)-I]
(a2- x2)-l cosh [b (a2_ x2)l]
O<x<a ~ n Jo [a (y2_ b2)~] 2
0 x>a
(a2 -x2)-!sinh [b(a 2-x2)!] (~ ntb! It {~ a [y + (y2 - b2)l]} X O<x<a
X Jig a [y- (y2_b2)!]} 0 x>a
(a2 - x2)-!cosh [b (a2-x2)!] (~ n)~b!J-l{~ a[y+(y2-b2)~J}x O<x<a
X J-ig a [y - (y2 - b2)!]} 0 x>a
sinh [b (a2 - x2)l] ~ nab (b2 - y2)-l II [a (b2- y2)~] 2
O<x<a O<ya ~ nab (y2 - b2) -l It [a (y2 - b2)~] y>b
2
0 x>a
O<x<a
x-~(a2 - x2)-! X
O<x<a
x-!(x2 - a2)-i X
sinh (ax) a - 1, b Rev < a
e-bx' cosh (a x)
sinh (ax) a <b ~--
x-£ e-a' x' sinh (a2 x2)
g(y) ~ ff(x) cos (x y) d x o
(+nyy!tLd~ a[b-(b2- y2)!J}X
X Ltg a[b+(b2- y2)!J}
--~------------
_ ~a(a2+y2)-I+ nb-1 x 2
sin (nab-1) cos (nyb-1) X cosh (2nyb 1) -cos (2nab 1) -~--~---- .. ~-------
r,'-2b-1 F(v+ 1) X
T[-t b-1(a-vb+iy)] } +i1fb-i (a+vb+iY)+1J
---
sin (2nab-1) X cosh (2nyb 1)-cos(2nab 1)-
(~ nYe-h '- : nyErfc(r~y)
y2 ,,1
n- i e - 16a' [2i a e- 16a' - y Erfc (2-~y a I)·
S. 65 -72. 1920.
S.65-72. 1920.
e-asinhx So,iY (a)
e-acoshx cos (b sinh x) cosh [Y arc tan (!)] Kiy [(a2 + b2)l]
: n(an)! [lHl-i Y) (a) Y-HHiy) (a) -
(sinh x)-~ e-2asinh .. - h(Hiy) (a) Yl(!-iy) (a) + li(Hiy) (a) X
X Y-!(!-iy) (a) - l-!(!-iy) (a) YHHi y) (a) ]
(cosh x)-l e-2acosh .. (:Y KHHiy) (a) KHl- iy) (a)
log (1 - e-2x) cosh x - : ny(y2+1)-ltanh(: n y) +
+(y2_1)(y2+1)-2
log (1 + cos a sech x) ny-1csch(n y) [COSh (~ ny) - cosh(a y)]
a<n
cosh x log (2 cosh x) - ~ n(y2+ 1)-lsech(~ n y) - x sinh x
log (tanh ax) - ~ny-ltanh (~) 2 4a
1 [ t+coshax ] 2n [sinh(:~)r og cos b+cosh ax
bs,n YSinh(n:)
ny-1sinh (b:) sech( ~~) b:;:;;'~n
2
b,d:;;;'n
(cosh x+ 1)"-
cos (tx2) + sin (tx2)
00 g(y) =fl(x)cos(xy)dx
ny-1csch - cosh - -cosh - a a a
r l n 2 [sech (n y)]2 P- Hiy (a)
~ nsech( ~ ny)[cos (~ ny2) - ~]
~ nsech(~ ny)[sin (~ ny2) + V~]
Siehe ERDELYI, A.: Tables of Integral Transforms, vol. 1, p. 36. New York 1954.
(~)! COS(ty2) + sin (ty2)
2 COSh(W y)
~ nsech( ~ n y) Lhy(a) + J-iy (a)]
~ ni CSCh( ~ ny) [J.y(a) - J-iy(a)]
cosh (~ n y) Kiy (a)
COShC ny)cos[~ Ylog(:~:)] X
X K;y [(b2 - a2)!] ab
(cosh x)-l sin (2a cosh x)
(cosh x)-! cos (2a cosh x)
(sinh x)-! sin (2 a sinh x)
(sinh x)-! cos (2a sinh x)
(cos x)-! sinh (2a cos x) .:n;
O<x<- 2
0 x>~ 2
O<x<~ 2
0 x>~ 2
O<x<n
0 x>n
O<x<n
0 x>n
X K iy [(b2 - a2)l] a b
- : n(an)![J_!(!_iy)(a) Y- HHiy) (a) + + J-HHiy) (a) L!(!-iy) (a)]
~ (na)! [Il(l-iy) (a) K HHiy) (a) + + I!(Hiy) (a) K Hi- iy) (a)]
~ (na)! [L!(Hiy) (a) Kk(}-iy) (a) + + L!(~_iY) (a) K!(Hiy) (a)]
~ n (n a)! I! (Hy) (a) I! (~_y) (a)
~ n (n a)! LH!-y) (a) L HHy) (a)
n(an)! cos (~ n y) Il(l-Y) (a) I HHy) (a)
n(an)!cos (~ n y) I _!(!_y) (a) I -!(Hy) (a)
§ 80 Orthogonale Polynome
[ Sinha] arc tan cosh (b x)
g(y) = jl(x) cos(xy) dx o
~ ny-1sin(a:)sechC nyb-1)
X).-l Pn (x) 0 < x < 1
o x>1 Re(A.+n) > 0
O<x<1 x>1
(a2 - x2)-l T2n (:)
x-l (1- x2)-l T,.(x) O<x<1
o x>1
-~nln(n+1) X 2
X y-i L!,nH(y)
1 1 1 1 1 1
2 '2 2'2'2 2
--n 1+-A.+-n --1 1 1 0 y2) 2' 2 2' 4
43
(a2+ X2)-!nX
(1- x2)-! cos [a (1- X2)!] X
X T;'n [(1 - X2)!] O<x<1
0 x>1
(1-x2)-!sin[a(1-x2)l] X
0 x>1
X 12n+l [b (a2 + X2)!]
sin[a(1- X2)!] U2,,(x) O<x<1
0 x>1
(a2+x2)-! X
(a2- X2Y-!C2n (:)
1 Rev> -2
(-1)n~n X 2
~n[(n _1)!]-lyn-l e-a y 2
(_1)n ~ n1~n[a(a2+y2)-~Jl2n[(a2+y2)!]
-
X T2n+l [a (a2 + y2)-!] 12n+l [(a2 + y2)!]
(-1)" (b2- y2)-!cos[a(b2- y2)!] X
X T2n [(1- y2b-2)] Y b
(-1)n(b2- y2)-!sin[a(b2- y2)!] X
X T2n+l[(1 - y2b-2)] Y b
(-1)"~na(a2+ y2)-! X 2
X U2,,[y(a2+ y2)-!] 12nH[(a2+ y2)i]
(-1)"a- 1 U2n (y) sin[a(1- y2)] O<y<1
0 y>1
§ 8. Orthogonale Polynome 45
Re'V> -1
X Pi~/l)(x) O<x<1
0 x>1
[(1 - x)O (1 + x)/l-
- (1 + x)V (1 - x)/l] X xR<v./l) (x) 210+1 O<x<1
0 x>1
e-~xl He2n (x)
e-lix' [Hen (X)]2
e-a x xo- 2n L2-;;:~ (a x)
Re'V> 2n-1
(-1)10 2°-! nl [(2n) !]-1 X
X r(2n + 'V -1) y-0-!J2n+oH(Y)
(-1)10 22n+0+/l [(2n) !]-1 X
X B(2n + 'V + 1, 2n +,u + 1) y2n X
X [eiy IF;. (2n + 'V + 1;
4n + 'V +,u + 2; - 2i y) +
+ e-iy 11\ (2n + 'V + 1;
4n+'V+,u+2;2iy)]
X B(2 n +'1' + 2, 2n +,u + 2) y2nH X
xi [eiY 11\ (2n + 'V + 2;
4n + 'V + It + 4; - 2i y) -
_e-iy 11\ (2n + 'V + 2;
4n+'V+,u+4;2iy)]
2
(-1)n( ~ nyy2n e-~Y'
(-1)mn!(~ ny y2m e-h 'L!m(y2)
~ (_1)10+1 r('V) [(2n - 1) !]-1 X 2
X y2n-l [(a - i y)-o - (a + i y)-V]
46 1. FOURIER-Kosin us-Transformationen
t(x) g(y) =ft(x)cos(xy)dx o
e-ax Xv- 2n- 1 L~-;;2n-l (a x) (-it ~ T(v) [(2n) !J-l X 2
Rev> 2n X y2n [(a + i y)-V + (a - i y)-vJ
e-!t x' Ln (x2) C ny (n!)-le- b ' [Hen (y)J2
x2m e-ltx' L~m (X2) (_1)m(~ nt(n!)-l e-h'x
X Hen (y) Hen+2m (y)
x2n e-1X' Ln-!t (~X2) n 2 (~ ny y2n e-itY' L~H (~ y2)
e-lx'[L;iC x2)r (~ nYe-b ' [L;i(~ y2)r
,-I·'L. (+.,) He,. (+ x) I (+ n)' ,-.,' L.(+ y') H,,. (: y)
e-!tx'L~C x2)L;!;-"(~ X2) I H-nYe- lty• L~(~ y2)L;,,-lC y2)
§ 9. Gamma- und RIEMANN-Zetafunktion
o <Rea <~ 2
g(y) = ft(x) cos (x y) d x o
2n-2 b-1 sech (:b) K (tanh :b)
22,,-3 [T(2a -1)J-l b-1 [cos (:b) r-2
0< y <nb o y> nb
§ 10. Fehlerintegral 47
Re (oc + fl) > - 1
-
X COS [~~ (f1- OC)] ° <y <nb
° y> nb
X [sinh(~)]i-fJ-l' ~~:-;;tfJ[COSh (;cx)]
~ ['If' (1 + ;n) - log ( ;n ) ]
n[y-l- ~csch(~ y)]
: n[cosh(~ y)+: #s(O,ie- 2Y)]
§ 10. Fehlerintegral
ex> I (x) g(y) = J I(x) cos (x y) dx
0
i e-a' x' Erf (i a x) Y'
1} __ -(y2) -n- a-Ie 4a' Ei - 2 4a2
i X-I e-a' x' Erf (i a x) - ~nErfc(~) 2 2a
X-l Erf [(a x)lJ logy -~log[a + (a2+ y2)iJ-
2
Erf(a x-~) y-l e-a V2Y sin (a V2y)
t(x) g(y) ~ rl(x) cos (x y) d x o
n- It a-l F. (1 ·l· _L) 1 1 '2' 4a2
Erfc (a x) y' . = - i y-le -4a' Erf (~)
y2 y2
X Erfc (a x) ~ a-2 e - 4a' - y-2 [1 - e - 4a'1
xv- l Erfc (a x) n- It a-Vv-l r( ~ + ~ v) X
Rev> 0 .E ( 1 1 1 1 1 y2 ) X22- V -+- v'- 1+- v'-- 2 '2 2' 2 ' 2' 4a2
x-l [Erfc (a x) - Erfc (b x) ] ~Ei( - L) - ~Ei( _ L) 2 4~ 2 4~
ea' x' Erfc (a x) - -n-~a-le 4a' El --1 _L . ( y2) 2 4a2
00 1 L (a2y2)n - 2n-"a X
Erfc(a X-l) n~O n!(2n+1)!
X [tp(2n + 2) + ~tp(n+1)-log(aY)1
Erfc [(a x)!] (! aY(a2+ y2)-! [(a2+ y2)~+ a]-!
x-l [1 - Erfc(ax!)] logy-~log[a + (a2+ y2)!]_
2
eax Erfc [(a x)!] (~: fit [a + y + (2ay)!]-l
Erfc{a [b +(b2+ x2)!J!} r! a e-ba'(a4 + y2)-! X
X [(a4 + y2)!+ a2]-!e-b (a'+Y')!
x-l Erfc (a x-!) - Ei[ - a (2iy)ft] - Ei[ - a(- 2iy)ftJ ----
Erfc (a cosh x) ~a-le-ia'W_i '1 (a2) 2 ,."y
eacosh' x Erfc (a cosh x) ~ sech (~ny) e!a' K1 (~a2) 2 2 '"y 2
C'll.\ll.x E~i a cosh x) 1 . _la' he) - n ~ e " sec - n y X 4 2 ,
§ 11. Exponentialintegral
§ 11. Exponentialintegral
e-ax' Ei (a x2)
Oherhettinger, Tabellen
+ ~ iEi(- ab+ia y)}
_ (a2+ y2)-1 g a log [(a+bi:+ y2 ] +
+ yarctan (a:b)}
_ (a2 + y2)-1 g a log [ (a-bi:+ y2] +
+ y arctan (a~b)}
- n y-l Erf (~ ya-!)
1 ~ 1 ~ E f C -tr) --:2n a-"e 4a rc -:2ya
y' 1 . ~ 1 -- (' 1 ) 22 n' a-" e 4a Erf 2 -:2 ya- i
1 -:2n2[csch(ny)]2[Iiy(a) +L;y(a)-
- e~:rcYJi (ia) - e-tr:rcy J_ i (ia)] y y
4
49
I (x) g(y) = fl(x)cos(xy)dx 0
si (a x) _ -.!..- y-l}og I y+a I 2 y-a y=j=a
_ -.!..- (b2+ y2)-1{-.!..- ylog[ b2+(y+a)2]_
e-bx si (a x) 2 2 b2+(y_a)2
- barctan (b2~;~y2) + nb}
~ y-l {2 sin (a y) Si(ub)+ Ci(ay + ab)-
Si(b x) O<x<a - Ci (I a y - a bl) + log I ~~: I} y=j=b
0 x> a -.!..- b-1[2 sin (a b) Si (a b) + Ci(2ab) - 2
- Y -log(2ab)J y=b
0 y> a
x(x2+ b2)-1 Si(ax) - ebY[Ei(- by) - Ei(- ab)J}
O<y<a
: n{e-bY[Ei(- ab) - Ei (a b)J} y>a
e-bx SiCa x) ~ (b2+ y2)-I{b arctan (at y )-
-b arc tan ( y-a)_~ylog[ b2+(y+a)2]) b 2 b2+(y_a)2
0 O<y<a Ci(a x) 1 -1
-2ny y>a
Ci(bx) O<x<a -.!..-y-l[2sin(ay) Ci (a b) - Si(ay + ab)- 2
0 x>a - Si(ay - ab)]
-.!..-nb-1cosh(by) Ei(- ab) 2
(X2 + b2)-1 Ci (a x) :nb-1{e-bY [Ei(ab) + Ei(- ab)-
- Ei(by)] + ebYEi(- by)} y~ a
§ 12. Integralsinus und Integralkosinus 51
t(x)
sin (a x) si(ax) + + cos (a x) Ci(ax)
X-I [Si (a x) cos (a x) -
- Ci(ax) sin(ax)J
cos (a x2) si (a x2) -
- sin (a x2) Ci (a x2)
cos (a x2) Ci (a x2) + + sin (a X2) si (a x2)
g(y) = ft(x) cos(xy) dx o
X {a log [(1 + a2~y2)2 + 4~:y21 +
+ 2 Y arc tan (_~ll y-)} a2+b2_ y2
~nlog I y+a I 4 y-a
1 (y2_ a2) _ny-l log --- 2 a2
: n y-qog(1 + :) ~ny-qog(y+a) 2 y-a
- ny-l [CU:) + s(::)]
B _1{' (y2)[ (y2) 1] n"{2a) ~ SIll -4a" S 4~- -2 +
+ cos ( ;': ) [C (:: ) - :]}
- sin (-::)[C (::) - {-J} 4*
X-I [Ci ( ;) sin ( ; ) -
o
-cos(acoshx)x : nsechC ny)SO,iy(a) X si (a cosh x)
cos (a cosh x) Ci (a cosh x) + + sin (a cosh x) X -: nycsch(: ny)S_I,iy(a)
X si (a cosh x)
§ 13. FRESNEL-Integrale
: (;y y-I(a2 _ y2)-l [a _ (a2 _ y2)l]!
1 y<a - - S(ax)
y>a
1 y<a - - C(ax)
y>a
0 y<a 1- C(ax) - S(ax) ~ a! y-I (y - a)-~ y>a
2
y-lsin(~) ' 4a
0 'V>1
X-I S (ax2)
X-I C(a X2)
cos (a X2) S (a X2) - - sin (a X2) C (a x2)
sin (a x2) S (a x2) + + cos (a x2) C (a x2)
[~ - S(ax2)] cos (a x2) -
- [~ - C (a X2)] X
X sin (a X2)
§ 13" FREsNEL-Integrale
1["(y2) "(y2)] 4 S1 ~ - C1 ~
- : [Ci U~) + si U~ ) ]
- sin (:;) Ci U~)}
- ~ (2na)-![sinU~)siU~)+
+[~ -s(ax2)]sin(ax2) - cos U; ) si U; ) ]
X-I [C (a X2) cos (a x2) + ~ n[~ - S(:~)] + S(ax2) sin(ax2)]
X-I [C (a X2) sin (a X2) - ~ n[~ -CU~)] - S(ax2) cos (a X2)]
Sea X-I) : y"-1 {sin [2 (a y)!] -
- cos [2 (a y)!] + e-2(a y)i}
53
I (x) g(y) = f/(x) cos(xy) dx o
C(ax-l) ~y-l{sin[2(ay)!] + 4 l + cos [2 (a y)l] _ e-2 (ay) }
~ - S(axl) 2
--n ay- cos --- h-1 l If ( a2 7 n ) (a2 )
4 8y 8 '" 8y
~ - C(ax!) 2
1 l f ( a2 5 n ) ( a2 ) --n ay- cos --- J-1- 4 8y 8 8y
nl r~ a y-I[cos (~- ~)h (~)- C(a x!) - S (a xl)
8y 8 '" 8y
n2-~sechC ny){sin(: + ~ a)x
sin (acosh2 x) C(acosh2x)- X cosh (n;) [~~ (;) + J-ii (;)] - - cos (a cosh2 x) X
X S (a cosh! x) -icos(: + ~ a) sinh (n;) X
x[~~(;) -J-ii (;)]}
nriisech(~ ny){cos(: + ~ a)x
cos (a cosh! x) C (a cosh! x) + X cosh (n;)[~i(;) +J_i~(;)] + + sin (a cosh2 x) X
+ i sin (: + ~ a) sinh (n;) X X S (a cosh2 x)
x[~~(;) -J_i~(;)]}
cos (a cosh x) C (a cosh x) + ~ n{ ~ sech C ny) [l\(~) (a) + H~ly(a)]-
+ sin (a cosh x) X -nasech(ny).[r(~ - ~ iY)X
X S(a cosh x) xr(~ + ~ iy)t S_!.iy(a)}
sin (a cosh x) C (a cosh x) - ~ n {~ sech (~ ny) [n.~) (a) + H~ly (a)] +
- cos (a cosh x) X + ~ nasech(ny) [r(! - ~ iY)X X S(a cosh x) 3 1 -1
xr -+-iy SlI,iy(a) 2 ) ] }
XC(2aCosh2 ;) + + sin (a cosh x) X
X S(2aCosh2 ;)
X C(2a cosh2 ;) -
X S(2a cosh2 ;)
g(y) = j,(x)cos(XY)dx o
! nsech(ny) {cosh (~ ny) X
X Lhy(a) + J-iy(a)] + + i sinh (~ n y) [.h y (a) - I-iy (a)]}
! nsech(ny) {coshC ny)x
X [J-iy(a) + .hy(a)]-
§ 14. LEGENDRE-Funktionen
o ~.(x)
-1 <Rev <0
Re.u> 0
(X2 1)tl'~~(x)
[x(1 + x)]-ll' ~~(1 + 2X)
g(y) =j,(x)cos(xy)dx o
- cos(~ nv)J.H(y)]
- ~ nl v(v+1) [re;V)r(1- ~ v)rx X y-l S i. H.(y)
21'+ln1 [r( -v2-'" )r( H;-"')rx X y-I'-! SI'-l .• H (y)
- ~ n~ YI'-l{J.H (~ y) X
X cos [~ y+ ~ (.u-v)] + YoH (~ y) X
X sin [~ y + ~ (.u - 'II) ]}
I(x) g(:v) = it (x) cos(x:v)dx o
0 O<x<1 - (~ ny yl'-l {1.H(Y) cos [: (p, - V)] + (x2-1)-Il'~~(x) x>1
1 + Y.H (y) sin [: (p, - v) ]} -- <Rep, < 1 2
Re p, > Re v > - 1 - Re p,
(1 - x2)-Il' P,,"(x) nI 21'-1 [r( 3-;+V )r( 2-;-V)r X O<x<1
0 x>1 X (p, + v) (p, - v -1) yl'-l LI'-i,.H (y)
Rep, < 1
x;'-l(1 - x2)-ll' P,,"(x) nl21'-;' rCA) [r( 1 + A-;+V) X
O<x<1 r( HA-p-V)r F. (~A HA. 0 x>1 X 2 23 2 '2'
ReA> 0, Rep, < 1 1 1+A-p-V A-p+v. y2) 1+ -~ 2' 2 ' 2' 4
~.(1 + X2) - 2ln-1sin(nv) [K.H (ri y)]2
-1 <Rev<O
(X2+a2+b2 ) ~. 2ab -1 <Rev<O
- 2 n-1 (a b)l sin (nv) K'H (a y) K.H (b y)
0----(X2+a2+b2 ) • 2ab
Q.L~2 -1) o <x <2a
X2 - ~ n2 a 1.H (a y) Y_._! (a y) O. -2-1 x> 2a ) Rev> -1 I
§ 14. LEGENDRE-Funktionen
0 x>a
(~+b2_~) ~v 2ab
0 O<x<a+b
~v (X2~:2~--'C) x>a+b
-1 <Rev <0
11: v 2ab
O<x<a-b
• 2ab
X-I ~v (2X2 -1) O<x<1
0 x>1
-1 < Rev <0
x-1 0.(1 + 2a2x-2)
g(y) ~r{(x)cos(xy)dx o
~ n(ab)![hH(bY)YvH(ay)-
- hH(a y) J-.-l,(b y)]
n (a b)! hH (b y) J-v-~ (a y)
- .!... n esc (nv) IF;, (v + 1; 1; i y) X 2
X IF;, (v + 1 ; 1 ; - i y) ~.--
1 2nr(1 +v) (ay)-IW_ v_!,o(ay) X
X [ctn(nv) M"H,o(ay)- - esc (nv) W_ v_!, o(a y)]
--.-~
+hH(~ bY)Y,;H(~ ay)]+
X [hH (~ a y) hH C by)
- YvH ( ~ a y) y';H (+ b y)]}
1
57
$. (cosh x) - : n-2 sin (nv) r(-V~iY)r(-V;iY) X
-1 <Rev <0 X r( 1 +V2+iY) r( 1 +V2-iY)
- 2v- i n-l a-! sin (nv) X
$o(a cosh x) X [cosh (n y) - cos (nv)]-1 X
-1 <Rev <0 xr( 1+V2+iY) r( 1 +V2-iY) X
a~1
x{p-o-~ (vact)+p-o-~ (-Va2 -1)} -l+'" a -!+." a
r(1 +v-,u+iY) r(1 +v-,u-iY) X (sinh x)p $~ (cosh x) n- i 2-1'-2 2 2
Re(1 + v -,u) > 0 r( -V-I') r(1 +v-,u) X
Re(v + It) < 0 X r( -V-:+iY) r( -V-:-iY)
Xr(t-,u)
X r(HV-:+iY) rC+V--;-iY)x
(a2 cosh2 x - 1 )!p X r(t-,u) r (1 +v-,u) X
X $~ (a cosh x) X r( -V-;-iY) r( -V-;+iY)
Re(,u+v) <0 X r(-v-,u) X Re(1 + v -,u) > 0
X F. C+v-,u+i Y 1+v-,u-iy. 2 1 2 ' 2 '
~ -,u; 1-a-2)
(a2 + b2 cosh2 X)-!{v+1) X r( 1 +v-,u+i Y) r( 1 +v-,u-i Y)
20 - 1 _I' 2 2 a X
X pI' [ b cosh x ] r(1-,u) r(1 +v-,u)
o (a2+b2cosh2x)t X yp-0-1 F. C +v-,u+iy
Re(v -,u + 1) < 0 2 1 2 '
1+v-,u-iy . 1 -,u" - a2 b-2 2 ' , )
§ 14. LEGENDRE-Funktionen 59
-1 < Re(v + tt) < 0
-1<a<1
-1<a<1
X [PHix(a) - P!.-ix(a)]
-1<a<1
X ['.l3Hix (a) - '.l3~-iX (a)]
a>1
-1<a<1
g(y) ~'J'j(x)cos(xy)dX o
xrC+V-:+iY)r(HV-:-i Y) X
X r (L+~±i+i Y) r (1 +V+:-i Y)
(~ Jl:y[r(~ -tt)r(sinO)f.tX
0 y> (j
00
(X-I L (-1t Cncos (n : y) Qn~ __ da) n~O ex 2
-(X~y~(X
i2fr[(a + coshy)!- (1 + coshy)!]
rlt n-1 (cosh y - a)-! X
1 l (cosh y+1)! + (cosh y-a)!, 1 X og 1 1
(cosh Y+ 1),'- (cosh y-a)"
a>1
a>1
coshy < a
o
X log r (cosh Y+ 1)!+ (cosh y-a)! j (cosh Y+ 1)!- (cosh y-a)!
X (a - cosh y) -,,-~
X 1.I3~!.~iX (a)
a> 1, Refh> 0
X [P~Hix(a) +
+P~Hix(- a)J
Q-Hix(COS b) + + Q-i-iX(COS b)
1 Refh> -
xl.l5,,_![(1- a2)-!coshyJ
ril nfr [r (: - fh) r1 (sinh a)" ein " X
X (cosh y - cosh ~)-,,-! y> a
o y<a ----------I~-------------------
- OW"~!r . (cosh a)J -"2"-~.x
X (cosh a + cosh y)-"
12n(a x)
].(a x)
§ 15. BESSEL-Funktionen vom Argument ~
O<y<a y>a
O<y<a
Rev> _ 1 - a· sin (: 'II n) (y2 -a2) -i X
X [y + (y2 - a2)lJ-v
y> a
O<y<a
: ['II ('II - 1)] -1 a cos [ ('II - 1) arc sin ( ; ) 1 +
+~ ['11('11 + 1)]-1 X 2
Rev> 1 ± ['II (V-1)]-l aV sin (±vn) X
X [y + (y2 - a2)lJl-v -
-: [v(v+1)]-laV+2sin(: vn)x
(-1t n1 (2a)-lP2n(;)
XV fv (a x)
t(x) g(y) ~rt(x)cos(xy)dX o
x 2R1 -+-v ---V' -'-( 1 1 1 1 1 y2) 4 2' 4 2' 2' a2
O<y<a
X[F(1+V)F(: - ~v)r(~rVx
X2r.1-V 1_- -v+- v+1 --D ( 1 1 1 3 . . a2 ) 2 -. 4' 2 4 ' , y2
y>a
n-kF(v+ ~)(2a)V(a2_y2)-V-Q
O<y<a
- ...!.. < Re V <...!.. I '( 1 ) 1 2 2 -n-fiF v+"2 (2 a)V sin (nv) (y2_ a2)-v-.
x-Vfv(ax)
O<y<a
o y> a
O<y<a
o y>a
21- vaV- 2[F(v)J-l R(1 i-v' ...!...1'2) 2 l' , 2' a2
O<y<a
y>a
I(x)
-1 <Rev < --.!.. 2
x (b2 + X2)-1 10 (a x)
x-V (b2 + X2)-1 ]" (a x)
Rev> _2 2
-1 <Rev <~ 2
0
O<y<a
2V+l:n:~aV[r(-v- ~)ry(y2-a2)-V-~ y>a
1 (a t21' "2 "2 F(v + ,u)[F(1 + v - ,u)]-1 X
( . 1 . y2) X 2F;.· ,u + V, ,u - V, "2' ~ O<y<a
(~ at r(2v + 2,u) [r(2v + 1)]-1 X
X cos [:n: (v + ,u) ] X
X y-2v-21' 2Fl (v + ,u, v + ,u + ~ ; 2 '
2v + 1; ;2) y>a
-.!..b-1:n:e-by Io (a b) 2
y>a
~ :n:b-v- 1e-by Iv (a b) y> a
~ :n:(-1tb2n-v-le-bYIv(ab) y>a
bV cosh (b y) Ky (a b) O<y<a
n=O,1,2, ... (-1tbY+ 2 ncosh(by) Kv(ab) O<y<a
-1 <Re(v+n) <~-n 2
I (x) g(y) = J I (x) cos(xy) ax 0
- (a2- y2)-i [log (a2_ y2)+y-log (~ab)] log (b x) fo (a x) O<y<a
- ...!...n(y2- a2)-i 2
XV sin x 1. (x) O<y<2
-1 <Rev <...!... nl2V-I[r(~ -v)r X 2
X [(y2+ 2y)-v-l_ (y2_ 2y)-v-iJ
2 <y < 00
x-V cos x 1. (x) nirv-i[r(~ +V)r(2y-y2)V-i
Rev> -...!... O<y<2 2 0 y>2
x-v sin x fv+! (x) ni2-V-I[r(~ + v)r (1 - y) (2y - y2)"-!
Rev> -.!.. O<y<2 2
0 y> 2
0 O<y<a X-I [si(ax) + ~ nlo(ax)] -nlog [(Y+a)l+(y-a)i] y>a
2yi
0 y> 2a
O<y<a-b 10 (a x) lo(bx)
n-l(ab)-i K {[ (a+:~:-YT}
§ 15. BESsEL-Funktionen vom Argument x 65
f(x) g(y) = ff(x) cos (x y) d x o
]. (a x) ]. (b x)
]. (b x) Lv (a x) a;z;.b
x-i [10 (a X)]2
Ip.(ax) ].(bx)
n-1 (a b)-l:O _, (a2+b2_ y2 ) v" 2ab
O<y<a-b
a-b<y<a-t-b
y>a-t-b
_a-1 P , --1 1 ( y2 ) 2 v-" 2a2 o <y <2a
0 y> 2a
O<y<a-b
a-b<y<a+b
C: Y {p-i [(1 - :~2)!lY y> 2a
WATSON, G. N.: J. London Math. Soc. Ed. 9, S. 21. 1936.
(: n yr~ (4a2 - y2)-!
o <y <2a
Rev> - : Xy-2v-l{2Fd! -t-v, : +v; 1 -t-v;
: -: (1- ::2)lJr Oberhettinger, Tabellen 5
[XV 1. (a X)]2
-~<Rev <~ 4 2
b> a, -1 <Rev <Re,u
x-·-'" 1. (a x) I", (a x)
Re ('II + ,u) > - 1
o < Rev < 2 + Re,u
Yo (a x)
g(y) = Jf(x)coS(XY)dx o
C nYr~ (4a2 - y2)-~COS [2'11 arccos (:a)] o <y <2a
o y> 2a
X (4a2 - y2)-' [n 1.l3=_l(4:2a~y2) _
- 2e- i "v sin (nv) O:-l (y:~:a2)]
o <y < 2a
o O<y<b-a
o
o y> 2a
O<y<b-a
BAILEY, W. N.: Proc. Lond. Math. Soc. Bd.40, S.37-48. 1936.
O<y<a y>a
f(x)
y'(ax)
+ 1. (a x) sin (~ 'I' n) -1 <Rev <1
XV [1. (a x) sin (a x) + + y'(ax) cos (a x)]
--.!...<Rev <-.!... 2 2
--.!...<Rev <-.!... 2 2
1.(ax)sin(ax- ~vn) -Yv(ax)X
g(y) = ff(x) cos (x y) dx o
-tan(~ vn)(a2_ y2)-A-x
X cos[varcsin(~)] 0 <y <a
- sin (~ 'I' n) (y2 - a2) -! X
x{a-V[y - (y2_ a2)!yctn(nv) + + aV [y - (y2_a2)!]-V esc (nv)} y> a
o O<y<a
y>a
O<y<a
o O<y<a
+ [y _ (y2 _ a2)!]'} y> a
o 0 <y < 2a
-.!...a-V(y2+ 2ay)-!x 2
+ [y + a - (y2+ 2ay)!]"}
X cos (: + ~ v n) - - Y,,(a x) X
X sin (: + ~ l' n) ] 3 3 --<Rev <- 2 2
log (b x) Yo (a x)
(b2 + X2)-l [~ n Yo (a x) -
- fo(ax) log x]
--~--I
I
Jo(a x) Yo(b x)
(y2 - a2) -} [r + log (y2 - a2) - log (~ a b)]
y>a
-Z(ny)-lK(Zay-l) y>Za
O<y<a-b
- n-l(ab)-~ K {[ y2-4~;~l~}
a-b<y<a+b
- 2n-1 [y2 - (a - b)2J-~ K{[ y2~~b_b)2 n y>a+b
f(x} g(y} = ff(x} cos (x y} d% o
o O<y<a-b
J.(bx) L.(ax)
].(ax) Y_.(bx)
b ! ( y2_ a2_b2 ) a::2; - n-l(ab)- cos (nv) O._! 2ab
Rev> -..!..- 2
a-b<y<a+b
y>a+b
O<y<a-b
a> b Rev>-..!..-, 2
a-b<y<a+b
4(ny)-l K[1 - 4a2y-2)l] y> 2a
2 n-1 [(a + b)2 _ y2]-i K{[ 4ab ]l} (a+b)2_ y2
O<y<a-b
n-1(a b)-i K{[ (a+4bl:- y2n a-b<y<a+b
4 n-1 [y2 _ (a _ b)2] -1 K{[ y2_ (a+b)2 ]i} y2_(a_b)2
y>a+b
O<y<a-b
a-b<y<a+b
1 [. ( y2_a2_b2) (a b) -lJ n-1 SIll (nv) 0.-1 2ab +
+ sec (n v) ~'-i ( y2-;,:2b- b2 )] Y > a + b
§ 16. BESsEL-Funktionen vom Argument x2 und 1/X
§ 16. BE55EL-Funktionen vom Argument;xil- und 1/JJ
f(x)
xl cos (a x2) 1-1 (a x2)
xl sin (a x2) 1-1 (a x2)
xl sin (x2) 1-1 (x2)
xi cos (x2) 1-~ (x2)
x [li(ax2)J2
g(y) =ff(x)cos(xy)dx o
- : a-l(ny)l[I-d:;) +H_1(:;)]
1 C Y -1 ( y2) "2 "2 ny a 1-14a
1 -2 C yJ (y2) "4 a Y"2 ny -1 4a
1 -2 C Y (y2) "4 a Y"2 ny h 4a
~ (~ ny)~ (a2+ b2)-le-laY'(al+b')-'x
-(ay)- cos ---1 ! ( y2 n) 2 Sa S
-- (ay)- sm ---1 1 . (y2 n) 2 Sa 8
l '9, ( y2 n ) ( y2) -n 2-'-YJ sin --- 1-1- 16 12 16
n 2- 6 y.ICOS --- 1-, -! H, (y2 n) (y2) 16 12 11 16
- : a-I [ 10 (;;) + Yo (:: ) ]
: a-I [10 (;;) - Yo (;; ) ]
1 C Y (y2) (yZ) __ a-1 -ny 1-1 - y; - 4 2 16a II 16a
1 -1 ( 1 l ] (y2) --a -ny _, - X 4 2 ~ 16a
71
t(x) g(y) = jt(x)cos(xy) dx o
y-l(~ nyrl[e-i~W.'_k(-i ;;)X
x~ I-k-v (a X2) I-Hv (a x2) ( ,y2) X W-.,-i - tsa +
in W (' y2) W (' y2) 1 + e 8" ., -1 t sa -v, -1 t sa
Yo (a x2) - ~ na-lY{Ild;~)r + [l-d;;)n
x~ Y-1 (a x2) 1 -1 e )! ( y2) -2"a 2"ny H_! 4a
xl! y!(ax2) 1 -2 e y (y2) -4 a y 2"ny H_I4a
x }i (a X2) Yi (a x2) - : a-I [10 (;;) + Ho (;; ) 1
xi 1_1(ax2) Y_ 1(ax2) 1 -1 C Y [J (y2) r -2"a 2"ny -1 16a _.
x [y!(ax2)J2 : a-I [10 (;;) - Yo U;) - 2Ho (;; ) ]
x [L1 (a x2) J2 -: a-1[1o(;;)+Yo(;;)+2Ho(;;)]
X-I J, (a x-I) 2 {J, [(2a i y)lJ K. [(2a i y)!J +
+ M( - 2a i y)lJ K. [( - 2a i y)!J}
nl4.l-2v r(A- v + ~) X X [r(i + 2v) rev - A)]-1 y2v-2.l-1 X
XU h.(a X-l) xoFa 2v+i, -+V-A,V-A; -- + (1 a2y2) 2 16
_1.. < Re A < Re v - -.!.- 4 2 + 4-.l-1a1+2.l r(v - A - ~) X
X[r(V+A+ ;W\Fa(~'A-V+;, 3 a2y2
A+V+-'- 2' 16 )
f(x)
Rev> - 2
Rev> -1
n = 0, 1, 2, ...
n = 0, 1, 2, ...
~ n],,(cyl) []"(dy!)cosC vn)
+ sinC vn)I.(c yl) K.(dyl)
c = 2[(a + W+ (a - WJ d = 2[(a + W- (a - WJ
a'2b
+ cosC vn)I.(cy!)K.(dy~)
c = 2 [(a + b)! + (a - WJ d = 2 [(a + b)~- (a - WJ
a;;:;;.b
x-l sin (a X-I) hn(bx-l ) I (-1t ~ nhn{2y![(a+W+ (a-b)Jl}x
n = 0, 1,2, ···1 Xhn{2y![(a + b)!- (a - WJ}
n = 0, 1, 2, ... X J2n+l {2 y! [(a-+ W + (a - b)iJ} X
xJ2n+l{2y![(a + b)!- (a - b)lJ}
-2<Rev<2
-1<Rev<1
o
+ :K.(cyl)sin(~ vn)]
d = 2 [(a + b)l- (a - b)!] a;;?;,b
- ~ nY,.(cyl)[.Tv(dyl)sin(~ vn)+
+Y,.(dyl)cos(~ vn)]
a~b
1 (x)
C> g(y) = Jf(x)cos(xy)dx
o
1 l -I [ . (a2 v :It) T ( a2 ) --n ay sm --- Jl( -1) - + 4 8y 4 • 8y
+cos --- fi,(+1)-( a2 V:lt ) ( a 2 )]
8y 4 • 8y
!(x)
xlv 1. (a x!)
-
g(y) = i!(x) cos(xy)dx o
( y t! ( a2 v n n) ( a2 ) - cos -----:4 -n 8y 4 4 v 8y
Z-vav y-V-l sin ( ~ _ ~ v n) 4y 2
b!v (a2 + y2) -Hv+1) e-ab (a'+y')-1 X
X cos [by(a2+ y2)-I_
- (v + 1) arc tan ( ~ ) ]
) 1 2y-l sm 4Y log 2Y -"2 C1 4Y +
+ ~ cos (:: ) si ( :: )}
(y2 _ a2)-! sin [: b2 y(y2 - a2J-l- nv] X
X J.. [: a b2 (y2 - a2)-1] y>a
y-l 1. (~) sin [~(a2+ b2) y-l_ ~ v n] v 2y 4 2
(n y) -1 {Ci (:: ) sin (:: ) -
- [n + si( ::) ] cos (:: )}
1 (n y [ (a2 ) ( a2 n) "2 y 10 8Y sin gy-4" +
+ Yo ( :: ) cos (:: - ;)]
1o(bx~) Yo (a x) + + 21o(ax) Yo(bxl)
10 (a xi) Yo (b xl) + + 10 (b x!) Yo (a xi)
1. (a xl) Y_ p (b xl) + + l_p(bx1) y"(axl)
-1 <Rev < 1
[x (a - x)]-lx
O<x<a x>a
Rev> -..!.. 2
g(y) = ft(x) cos (x y) dx o
(a2 - y2)-l cos [: b2 y (a2 - y2)-1] X
X Yo [: ab2(a2- y2)-1] y < a
(y2_ a2)-!sin[: b2y(y2_a2)-1] X
XYo[: ab2(y2_a2)-1] y> a
-1[' (a2+b2) y; (ab ) (a2+b2) T (ab)] y sm -- 0 - -cos -- Jo - 4y 2y 4y 2y
y-l sm -nv+-- Y - -[. (1 a2 + b2 ) ( a b ) 2 4y p 2y
( 1 a2+b2 ) ( ab)] -COS 2: nv +4y- 1. 2Y
n1.{: a[(b2+y2)l+YJ}X
X1.{: a[(b2+y2)!_YJ}cos(: ay)
o O<x<a (1 ) -2 b p y-P-lu,,+1(2ay, ba!)
x>a I Rev> -1
_________ 1 ______________ _
Rev>...!... 2
Rev> -...!... 2
- cos (~ y - n v) 1. (Z2) Y" (Zl) ]
Zl =...!... [y + (y2_ b2)t] 4
Z2 =...!... [y - (y2_ b2)lJ 4
y>b
(:Y (~ ab)" cos (~ a y) (b2- y2)-!(V+l) X
XKpH [ ~ a(b2- y2)!] y -1
Rev<~ 2
Rev> -~ 2
(b2 _ y2) -! e-ia (b'-y,)l X
X cos {v arc tan [y (b2 - y2)-i] - ~ a y}
y b
o O<yb
- (~ naY (a bY (b2 - y2) -~ (vH) X
xY.H[a(b2-y2)i] 0 <y b
(~ nat(ab)-V(b2_y2)~(V-!)X
o O<y _~ y> c 2
10 (b (a2 - X2)!i] O<x<a 0 x>a
(a2 - x2) -~ 1. [b (a2 - x2)lJ
O<x<a
(a2 - x2)lv 1. [b (a2 - x2)lJ
O<x<a 0 x>a
Rev> -1
0 O<x<a
x>a
x>a Rev> -1
o O<x<1
x(x2-1)lv 1.[a(x2-1)~J
-~--"-----
X .4v g a [(b2 + y2)li - y J}
(~ anY (ab)V (b2+ y2)-HvH) X
x1.H[a(b2+ y2)lJ
(b2 _ y2) -l e-a(b._y.)k O<yb
(:~)~ (a b)" (b2 - y2) -k(vH) X
X KvH [a (b2 - y2)lJ O<yb
- ~ n.4 vg a[y - (y2_ b2)lJ} X
X Y- lv g a [y + (y2 - b2)lJ} y> b
o O<y<a
§ 17. BESsEL-Funktionen vom Argument (ax2 +bx+c)i 79
f(x)
X1.[a(x2- c2)i] X> c
-1 <Rev<2- 2
o O<x<a
x>a
Rev> -1
O<y<a
O<y b
x>a
X 1.+2n [(a2 + x2)l] n = 0, 1, 2, ...
- :n;-I(b2- y2)-!{sinh[a(b2- y2)l] X
X [Ci (zl)+Ci (Z2)] - cosh [a (b2 - y2)!] X
X [Si(ZI} - Si(Z2)J} 0 < y b
(-1t (~ :n;at a-'(1 - y2)i('-!) X
X C~n (y) 1.-! [a (1 - y2)!] 0 < y < 1
Rev> _2- 0 2
0<y<1 n = 0, 1, 2,... 0 y>1
o
n=O'i'2"~'I' Rev> --
X C;!~ [y (a2 +y2) -Q] h+2nH [(a2 +y2)~]
1
- K!v{ ~ a [y + (y2_ b2)!]} X
O<y b
C naY (a b) -v (b2 - y2)Hv-,}) X
(a2+ x2)- It V Yv[b(a2+ X2)~] X Y._i[a(b2- y2)!i] 0 < y - ~ _(~ay(ab)-V(y2_b2)!t(V-~)X
o x> a
n-1 (b2 + y2) -Ii {sin Cl [Ci (Zl) + Ci (Z2)] -
-- cos Cl [Si (Zl) + Si (Z2)J)
Cl=a(b2+y2)!i, zl=Cl+ay, z2=Cl-ay
§ 17. BESsEL-Funktionen vom Argument (ax2+bx+c)i 81
/(x)
-1 <Rev <1
x> a
o O<x<a Yo [b (X2 - a2)!] x > a
o x(b2 + X2)-lX
X y,; [a (X2- c2)!] x> c
n=O,1,2, ... 1 "
: n{cos(: n'll)[.4.(Zl)Yi.(Z2)+
X [,4. (Zl) ,4. (Z2) + Yi. (Zl) Yi. (Z2)] }
Z1 = ~ a [(b2+ y2)! + y] 2
Z2 = ~ a [(b2+ y2)~ - y] 2
- ..!. n sec (~'11 n) [1!. (Z1) 1!. (Z2) + 4 ,2
+ 2 cos ('II n) h. (Z1) .4. (Z2)]
Z1 = ~a[y + (y2_ b2)!] 2
Z2 = ~ a [y - (y2 - b2)1] 2
y>b
- n-1 (b2 - y2) -i {sinh [a (b 2 - y2)!] X
X [Ci (Z1) +Ci (zz)] - cosh [a (b2 - y2)!] X
X [Si(Z1) - Si(Z2)J} 0 <y b
Zl = a [y + (y2 - b2)!]
Z2= a[y - (y2_ b2)l]
arg[(y2- a2)l] = ~~ fur y~:
(- 1)n+1 (b2 + c2)~.+n-! cosh (b y) X
xK.[a(b2+ c2)!] 0 < y < a
6
x~1-1g a [(b2+ X2)~- e r~ -- b]}l-!g X
"2 n- y (a2 - y2) -! cos [b (a2 - y2)!]
O<y<a
Xa[(b2+x2)t+b]} 0 y>a
1. {a [(b2+ X2)~+ xJ}x (4a2 - y2)-! h. [b (4a2 - y2)!] O<y<Za
xl.{a[(b2+x2)!- xJ}
Rev> -1 0 y> Za
1. {a [x + (xZ-b2)!J}x (4a2 - y2)-t 12• [b (4aZ - yZ)t] O<y<Za xl.{a[x - (X2- b2)!J} 0 y> Za
xl 1--1 g a [(b2 + X2)!- (~ n- y r! (a2 - y2) -! sin [b (a2 - y2)t]
-bJ}y-1g X O<y<a
( 1 ) -! (2 2) -! -b(Y'-a')~ xa[(b2+ X2)~+ b]} - "2n-y y - a e y>a
Y,;{a [(b2+ x2)l+ xJ}x - (4a2- y2)-! h.[b(4a2- y2)!]
o <y <Za xY,;{a[(b2+ x2)!- xJ}
4n--1 cos (n-v) (y2_ 4a2)-! K 2,. [b (y2_ 4a2)!J -1 <Rev<1
y> Za
1. (Zl) y" (zz) + 1. (zz) y';(Zl) Z (4aZ - y2)-! Yz. [b (4aZ - y2)}]
o <y <Za Zl = a[(b2+ x2)l+ xJ -1 . 2 2 _1 2 Z
§ 18. BESSEL-Funktionen mit trigonometrischem und hyperbolischem Argument
o
!(x)
Rev> - ~ I
n- l.-y (a) l.+y (a)
§ 18. BESsEL-Funktionen mit trigonometrischem und hyperbolischemArgument 83
I(x)
Y2 v [2 a cos (~ x) 1 O<x<J1
0 x> n
-~ <Rev <~ 2 2
0 x>n Rev> -1
y"(2a sin x) O<x<n 0 x>n
-1 < Rev < 1
2
Y2• [ 2 a cosh (~ x) 1
fo [a (2 sinh x)~]
t To [a (2 cosh x) ]
g(y) =JI(x)cos(xy)dx o
n csc (2 nv) [cos (2 nv) ],,+y (a) ],,_y (a) -
- h-.(a) f-y-v(a)]
ncsc(nv)cos(~ ny)x
- I-~(v+y) (a) I-!(v-y) (a)]
[Iiy (a) + L iy (a)] K iy (a)
I v- iy (a) Kv+iy (a) + I v+iy (a) Kv- iy (a)
- 2n-1cosh(ny) [Kiy(a)]2
csc (2 nv) {cos (2 nv) [Iv- iy (a) Kv+iy (a) + + I v+iy (a) Kv- iy (a)] - K v- iy (a) X
xLv_iy(a) - LV+iy(a) KV+iy(a)}
1 - "2 n [],,+iy (a) y"-iY (a) ],,-iv (a) y"+iy (a)]
1 "2 n [],,+iy (a) ],,-iy (a) - Y.+iy (a) Y.-iy (a)]
[];y (a) + I-iy (a)] K iy (a)
i n-1[Kiy(a ei~) Kiy(a ei34'!) -
6*
1';, [a (2 cosh x)i]
g(y) = j/(x) cos(xy) dx o
- n-1 [K.,,{a e'~) K. y (a ei3t ) + + Kiy(ae-i~) K.y(a e-i3t )]
- i !!.- H.<2) (a) H(2). (b) 2 oy -oy
= - i!!.- e~ y HJ2) (a) HJ2) (b) 2 .y oy
§ 19. BESSEL-Funktionen mit variabler Ordnung
I (x)
Rev> -~ 2
csch(~ nx)x x [I. x (a) - 1-.x(a)]
X [~x(a) + Y_ ... (a)]
- 2i cos (a coshy)
f(x)
sech(nx) sech(~ x) X
sech (nx) sech(~ x) X
X [Y;x(a) + Y_ix(a)]
+ S(2acosh2 ~)]
-icos(acoshy) [C(2acosh2 ~)+
+ S ( 2 a cosh 2 ~)] + + i sin (a coshy) [C (2a cosh2 ~)-
- S (2 a cosh 2 ~)]
2 cos (a cosh y) [C (2a cosh2 ~)-
- S (2 a cosh 2 ~)] + + 2 sin (a cosh y) [C(2a cosh2 ~) +
+ S (2a cosh2 ~) -1] 2icos(acoshy) [C(2acosh2 ;)+
+ S(2acosh2 ~) -1]-
- 2 i sin (a cosh y) [ C (2 a cosh 2 ~)-
- S (2 a cosh 2 ~)]
2cos(acoshy) [C(2acosh2 ~)+
+ S(2acosh2 ~)-1]
- 2 sin (a cosh y) [c ( 2 a cosh 2 ~) -
- S (2 a cosh2 ~)]
sech(nx)csch(~ x)x
hx(a) Lix(b) + + I-;,,(b) ¥;,,(a)
H/!) (a) ~~) (h-) e""
-C(2acosh2 ~)]
- S(2acosh2 ~)]
- fo[(a2+ b2+ 2abcoshy)l]
Yo [(a2+ b2+ 2abcoshy)~]
I(x) g(y) = f/(x) cos (x y) dx o
r! [(b2 - a2 - y2)2 + 4b2 y2]-! X
b:2:;a x{[(b2 - a2- y2)2+ 4b2y2]k+
-l--b2 - a2 _ y2}~
I(x)
Ko(ax)
K.(ax)
sinh (: ax) Kl (: ax)
-~<Rev <~ 2 2
TIMPE, A.: Math. Ann. Bd. 71, S. 480- 509. 1912.
~n(a2+ y2)-~ 2
+ a'[Y + (a2+ y2)lJ-.}
: nl(2a)·r(: +'1') (y2+ a2)-.-!
11(1)"rC 1 +1) - a- - a - v - - f1 - X 2.2 2 2 2
Xr ---f1--v Jl-v--f1+-C 1 1) C 1 1 2 2 2 2 2 2'
1 1 1. 1 . 2 -2) 2-2v-2f1, 2' -y a
~n(a2+ y2)-i X 2
na2(2y)-! (a2+ y2)-! [y + (a2+ y2)kJ-l!
: nll[r(: +v)tsec(nv)a-'x
[y2+ (a + b)2J-l K{ 2(ab)l } [y2+(a+W]l
~ (ab)-lO _, (a2+b2+ y2 ) 2 ... 2ab
88 I. FOURIER-Kosin us-Transformationen
t(x) g(y) = ft(x)cos(xy)dx o
x!I_1(ax) Ki(ax) (2: Y (y2 + 4a2)-i
X! Iv- 1 (a x) KvH (a x) a- 2V (2: y (y2 + 4a2)-~ [(4a2 + y2)! _ yJ2v
Rev> -1- 4
x-It Iv (a x) K. (b x) ein '(2: r r(! +1') [r(: -v)r X
Rev> -...!..- X O=Hy-l (y2 + 4a2)ltJ X 4
X s;j3=i [y-l (y2 + 4a2)!J
K.(a x) K.(b x) ...!..- n2 sec (nv) (a b)-It F._1 (a2+b2+~)
-...!..-<Rev<...!..- 4 ~ 2ab 2 2
Ko (a x) Ko(b x) [Ko (c X)]-1 OLLENDORF, F.: Arch. Elektrotechnik Ed. 17, S.79-101. 1926.
Ko(ax) Ko(bx) n [y2 + (a + b)2J-! K [( y2+ (a-b)2 )!] y2+(a+b)2
n z-v-.u-lJ r(1 + 21') [r(1 + ,u)]-1a.u-v-lJ X
x·-.u !.u (a x) Kv (b x) X y~ (cosh {} - cos <5) F.=~-! (cos <5) X
Re,u> -1, Rev> -...!..- X (sinh {})-! cosh [(v -,u) {}J 2
+'b' C .1) Y ~ = ~ a ctn 2 <5 + ~ 2 {}
x·-.u I.u (a x) Kv (b x) 00
Rev> -...!..- a> b ~ n J t·-.u !.u(at)],,(bt)e-tYdt
2 ' 0
x-2• I. (a x) K. (a x) n! Z-2.-1 re - v) [r(1 + V)J-l y2v-l X
1 1 --<Rev <- e 1 ' 2 2 X 21\ 2' 2 -v; 1 +1'; - 4a2y-2)
X2.+1 Iv (a x) Kv (a x) _...!..- a-I sin (nv) y-2.-1 X 2
-...!..- < Rev <0 2 X 2 1 - - + v' - v· - - y2 a-2 R C 1 1 ) 2' 2 ' , 4
§ 21. Modifizierte BESsEL-Funktionen yom Argument x2 und 1/X 89
!(x)
x-~[K.(ax)]2
-~<Rey <~ 4 4
-2<Rey <~ 4 4
Re A > IRe,u I + I Re y I
g(y) = ff(x)cos(xy)dx o
X (y2 + 4a2) -~ O=i [y-1 (y2 + 4a2)!] X
X O=i [y-1 (y2 + 4a2)!]
ei 2". r (: + Y) [r (: - Y) r X
X (2: )'" {0=1 [y-1 (y2 + 4a2)!.J}2
- e2i"'(~ ny)"'rU +y)x
2).-3 [r(J.)]-l r( A+~+V) rC'+~-V) X
xr( A-~+V)r( A-~-V) X
X R (A+tt+V A+tt-V A-tt+V 43 2 ' 2 ' 2 '
A-tt-V.1 A 1+A._ 1 2) 2 '2' 2' -2-' 4"Y
§ 21. Modifizierte BESSEL-Funktionen vom Argument W und 1!i£
!(x) g(y) = ff(x) cos (x y) dx o
1 1 _y-,- -(aY)-"'e Sa 2
Rev> -~ 2
-~<Rev <~ 4 2
Ko(a X2)
eX' KO(X2)
-~<Rev <~ 2 2
g(y) ~'!t(x)cos(xy)dx o
C )-~ n-lZ-V-~r(-v)r2+2ve 8X
( 1 . . y2) X IF;. 2 - v, 1 + v, 8 + y'
+ 22v-~ y-2v e- T r(v) X
x[r(: -v)t\F;.(: -2V;1-V;~2)
22V-!a-!r(: +v) [r(-v)]-lx
xy-2v-l e-Ta F, ---2V'-V'~ y' ( 1 2 ) 1 1 2 ' , Sa
~na-lyK, (~) [Ii (~) +L, (~)l S "Sa 4 Sa " Sa
1C t y' (y2) 2 2 n " e1G Ko 16
C )~ y' ( y2) 2 n e--16 I o 16
1 -2 C Y [I ( 1 2 2) 4 n a 2 n y -1 4 y a- -
- L-i (: y2 a-2) 1
n ( )-v-! r(H2V) -l- 2 2a r(1 +v) e a X
X IF;. - - V; 1 + V;-( 1 y2) 2 Sa.
nz-!vr(~ -2V)[r(~ +v)ryv-1X
§ 22. Modifizierte BESsEL-Funktionen vom Argument (ax2 + bx + e)l 91
t(x)
ReA> IRevl
xi I -k-' (a2 x2) K i -. (a2 x2)
Rev < ~
x 2F2 A+V,A-V,-,A+-,-~ ( . 1 1 . y2) 2 2 S
~a-2(-'!JI-y I (~)K (~) 4 2 -! 16a2 , k 16a2
[r ( ! ) r r (~ - v) y-1 ( 2yn y X
xU;:, -i( S~2) M_., -i (S:22)
- n Lay y {Jo[(2ay)l] KI[(2ay)!] +
+ .h [(2a y)l] Ko [(2a y)!J}
- n Ko [(2a y)!] Yo [(2a y)l]
- nK.[(2aY)!]{L[(2ay)~]sin(~ vn)+
+ Y,;[(2ay)!] cos(~ vn)}
t(x)
x-! K.(a x!)
-1 <Rev <1
g(y) =ft(x)cos(Xy)dx o
[ . ( a2 ) . (a2 ) . ( a2 ) ( a2 ) 1 (2 y)-I C1 4Y SIll 4Y - Sl 4Y cos 4Y
1 (1 )(n)~ - "4 nsec zvn y X
X h - sin ----- + [ ( a2 ) ( n v n a2 ) "v Sy 4 4 Sy
( a2 ) ( :r; v n a2 ) 1 + Y! - cos -' ----- v 8y 4 4 Sy
1. (a X!) K. (a x!) ~ ncsc(nv) y-l[sin(~ nv)1.(;;)+
Rev> -1 1 . J (" a2 ) 1. J ( . a2 )] + -~ ~- --~ -~- 2 • 2y 2' 2y
Yo (a xl) Ko(ax!) t -1 K (a2 ) --y 0-
2 2y
[1.(axl)sin(~ vn)+
XK.(axi )
-1 <Rev < 1
x-i 1. (a xl) K. (a xl) a-2r(: + ~ v) [r(1 +V)]-IX
Rev> -~ 2 C Y w, ( a2
) M ( a2 ) X "2 ny i.l. 2"Y -U· 2"Y
x-~ K. (a xl) X x[cos(~ vn- ;)X X 1. (a xi) +
+cos(~ vn+ ;)X - a-2(~n yw (~) w, (~) 2 Y -i.i· 2y t.!· 2y
X Y,,{axi)]
-~<Rev<~ 2 2
x ~ (xt ) Kl (xl) t y-3K e -1) -"4 o"2 Y
10 {a xl) Ko(axl) : n y-l[COS (;; )1~(;;) +sin (;;) Yo(;;)]
[I. {a xl) + L.(axl)] X ~ny-l[cos(~vn -~) 1.(~)- l 2 2 2y 2y xK.(ax)
-1 < Rev < 1 . (1 a2 ) ( a2 )] -SIll "2vn -2"Y Yo 2"Y
§ 22. Modifizierte BESsEL-Funktionen vom Argument (ax2 + bx + c)! 93
t(x)
x!'{ i"f K. [a (i x)!] + + e-;"T K.[a(- ix)itJ}
Rev> -1
x-itK.Ca(ix)~]x
xK.Ca(- ix)!]
K. C a (i x)~] K. C a (- i x)!]
-1 < Rev < 1
- ~ nY.(axit)]
g(y) = ft(x) cos(xy) dx o
a' n 2-v- l a2 • y-v-l e- 4Y
1 2(1 yrC 1 )rC 1) 2 a- 2 ny 4+2 V 4-2 v X
xW; i- W; -i-( a2 ) ( a2
) •. it· 2y •• !. 2y
;; [Ho (:: ) - Yo (:: ) ]
1 -1 e ) 5 (a2 ) - n y sec - n v 0 • -
4 2 • 2y
( a2 ) ( a bit) n y-l cos 4Y log 2Y
1 (1 r' 1 ( a2 1 ) -n -a y'- cos ---vn 2 2 4y 2
2 -v .-1 ( a 1 ) Y sm ---vn
4y 2
(x2+ a2)Hv Kv[b(a2+ X2)t]
[x(i + x)]-!x
X K!H b [(a2+x2)~+a]}
Io{a[(b2+x2)!- xJ}x
xKo{a[(b2+ X2)!+ xJ}
I.{a[(b2+ x2)l_ xJ}x
xK.{ a [(b2+ X2)! + xJ}
Ko{a[x + (x2- b2)tJ}x
xKo{a[x - (X2- b2)!J}
g(y) = ff(x) cos(x y) d x o
~K1V{~a[(y2+ b2)!- y]}x 2 " 2 X K!vg a[(y2+ b2)}+ y]}
UtaY (a b)'!'· (b2+ y2)H.-! X
X K ±v-t [a (b2 + y2)!]
+n2sec (~ v n) {cos (~ y) [hV(Zl) hV(Z2)-I-
+ l!v(Zl) l!v(Z2) 1 -I- sin (~ y) X
X [h v (Z2) l!v (Zl) - h. (Zl) l!v (Z2)]}
Zl = ~ [(b2+ y2)!+ y] 4
Z2 = ~ [(b2 + y2)l_ y] 4
(2: Y (b2 + y2)-l e-a(b.+y.)t
~ n(4a2+ y2)-!{Io[b(4a2+ y2)i] - 2
- Lo [0 (4a2 + y2)lJ}
xI2.[b(y2+ 4a2)l] + i ~ ncsc(2nv) X
X (J2 • [i b (y2 + 4a2)t] -
-J_ 2.[ib(y2+ 4a2)l])}
- Yo [b (4a2 + y2)lJ}
!(x)
xKv{a[(b2+ X2)lr+ xJ}
(a2 - x2) -~ Iv [b (a2 - X2)!]
O<x<a
o x> a
O<x<a o
I o[b(a2- X2)lr]
(a2 - X2)~V Iv [b (a2 - X2)lr]
O<x<a 0 x>a
Rev> -1
(y2_ b2)-lsin[a(y2- b2)!]
(~ ant(aW(b2_y2)-HVH) X
(~ anY(ab)"(y2_b2)-lr(VH) X
y>b
- ~ (y2_ b2)-~{sinlX[Ci(zl) + Ci(Z2)] -
Ko [b (a2 - x2)l] 0 < x < a - cos IX [Si (Zl) - Si (Z2) J}
o x>a lX=a(y2-b2)~, zl=ay+lX, z2=ay-1X
arg(y2-b2lr= 0 fur y~b in
-1 <Rev<1
-1 < Rev < 1
o O<x<a Ko [b (X2 - a2)1] x> a
(a2- x2)-l].[b(a2 - x2)IJ
-1<Rev<1
O<x<a
- 2n-1 (x2 - a2)!vx
I
o
+ Yi,(Zl) L!.(Z2)] = - ~ n2 X
+ h.(Z2) Y-!.(Zl)]
~ n2secC 'lin) [h,(Zl) h.(Z2) +
~ (b2 + y2) -! { sin oc [Ci (Zl) + Ci (Z2) ] -
- cos oc [si(zJ + si(z2)J}
oc = a (b2 + y2)1, Zl = a Y + OC, Z2 = oc - a y
- ~ nYi.g a[(b2+y2)i+ YJ}X
Xy1.g a[(b2+y2)i_ yJ}
(~ naY (a W (b2 + y2)-iCv+!) X
X y"H [a (b2 + y2)!J
§ 23, Modifizierte BESsEL-Funktionen 97
CX>
12v[2acos(~ x)] O<x<n
0 x>n n I._ y (a) I.+ y (a)
Rev> --.!.. 2
K2v[2acos(~ x)]
O<x<n ncsc (2 nv){Iy_. (a) Ky+.(a) sin [n(y+v)]-
0 x>n - Iy+.(a) Ky_.(a) sin [n(y-v)J}
--.!..<Rev<-.!.. 2 2
0 x>n ncos(~ ny)I.-b(a)I.Hy(a)
Rev> --.!.. 2
K 2• (2 a sin x) O<x<n ~ n 2 csc(2nv) cos (~ n y) [L.-b(a) X
0 x>n
Ko[2asinh(~ x)] : n2{[,hy(a)]2+ [~y(a)]2}
e-a' cos' x Ii. (a2 cos2 x) r!·-ln!a· rC~V) [r(2+;-Y) X O<x<~
2 r(2+V+Y)t F. C+V 1+v 2+V; x>~
X 2 33 2' 2' 2 0
2 2+v-y 2+v+y 1 + 'II' _ 2a2) Rev> -1 2 ' 2' ,
K2.[2asinh(~ x)] : n2{kv_.(a) ,hy+.(a) +
--.!..<Rev<-.!.. + ~y_. (a) ~y+. (a) + tan ('II n) X
2 2 X [,hy+.(a) ~y_.(a)- ,hy_,(a) ~y+,(a)J} Oberhettinger, Tabellen 7
98 I. FouRIER-Kosinus-Transformationen
2n-1 K o [a (2 sinh x)!] -
- Yo [a (2 sinh x)l]
Ko[(a2+ b2+2abcoshx)!]
o
K.+i'Y (a) K.- iy (a)
Kiy (a i'l) Kiy (a e-i'l)
- [Y;y (a) + Y- iy (a)] Kiy (a)
Kiy (a) Kiy (b)
!(x)
sinh (b~ [I . (a) - I· (a)] cosh (nx) -u u
= 2i n-1 tanh (nx) X
X sinh (bx) Kix(a)
b~~ -2
g(y) = J!(x) COs(xy) dx o
MAcRoBERT, T. M.: Proc. Roy. Soc. Ed., Bd. 55, S. 87. 1934.
~ 12 • [2 a cos (~ y)] O<y<n
o y> n
_ e-acosh(y+ib) Erf [i (2a)t sinh (Y~ib)])
_ ~ i {e-aCOSh(Y+ib) X
+ e-acosh (y-ib) Erf [i (2 a)! cosh (Y~ib l])
§ 24. Modifizierte BESsEL-Funktionen mit variabler Ordnung 99
{(x)
C ' sech 2"nx)K;x(a)
b~.l. n -2
b;:;'~n 2
[hx(a) + I-;x(a)] K;x(a)
[Y;x(a) + L;x(a)] K;x(a)
sech (nx) [Ii x (a) + + L;x(a)] K;x(a)
[K;x(a)]2
K ix (a) K ix (b)
K.+;x (a) K.- ix (a)
~ n e-acoshy 2
~ n cos (a sin b sinh y) e-acosbcosh y 2
, 1 n eacosh y Erfc [(2a)~ cosh (1 y) 1 - - 2 2
00
: n{eacOSh(Y+;b) Erfc[(2a)1cosh(Y~ib)1 +
+ eacosh (y-ib) Erfc [(2a)~ cosh (Y~ib)]}
00
~ n2b-1 L (-1r En In!'Ja) cosh(nn n n=O b
~ n 10 [a (2 sinh y)~] 2
~ n Yo [a (2 sinh y)!] - Ko [a (2 sinh y)!] 2
~ n Jo [ 2 a sinh (: y) 1
: n [10 ( 2 a cosh ;) - Lo (2 a cosh ;) 1
~ nKo[2acoshC y)]
~nKo[(a2+ b2+ 2abcoshy)1] 2
: n K 2. [ 2 a cosh (~ y)]
7*
x' 51',' (a x) Rev> -~
2
(a2 + x2)-l 50" [b (a2 + x2)lJ
(a2+ x2)-1 X
X g n sec (~ v n) X
X I. [b (a2 + x2)lJ + + i so,. [i b (a2 + x2)lJ}
50,ix (a)
o
C)! C 1 1) 2"n 2"'r 2" + 2",u +2"'1' X
O<y<1
p'-p-i ( ) X ._! Y O<y<1
0 y>1
~(~y r p r( -~v- ~,u) X 4 2a 2 2
X r(~ '1'- ~ ,u) (y2_ a2)HpH) X
\l3",H (Y) X '-l a
n1(2v + 1)-12·+1' a' r(1 + ~ ,u + ~ v) X
[(1 1 1)r C X r2"-2",u-2"v :A 2"+'1',
1 1 + 1 3 + ' 1 2 -2) ---,u -'1"- V -ay 2 2 2' 2 '
~ K~.g a[y + (y2_ b2)lJ} X
X Kl.g a[y - (y2_ b2)lJ}
~ nIl.g a[(b2+y2)l_ yJ} X
X Kk.g a[(b2+y2)l+YJ}
e-asinhy
§ 26. ANGER-WEBER-Funktionen 101
r( 1 1 1 . ) X ---ft+-ZX X
222
XS,..,ix(a)
Reft <"2
g(y) =ff(x)cos(xy)dx o
- cos(acoshy) Ci(a cosh y) - - sin (a cosh y) si (a cosh y)
(2a)-l r,..-l X
xr(: - ~ ft+ ~ iy)S,..H,iy(a)
x1v[b(a2+ x2)!J + + i esc (nv) X
X [Jv(ibVa2+ X2)-
2 cos (~ nv) (a2 - y2)-i cos [v arc cos (~)]
O<y<a
y>a
~ n i csch (n y) [h(V+i Y) (a) J-Hv-iy) (a) X
xcos G v + i ~ y) - h(v-iy) (a) X
X J-~(.+iy) (a) cos (~ v - i ~ y)]
~ i:re csch (:re y) [l-H.-iY) (a) IH.+iy) (a) X
E.(2acoshx) xsin(;v+i; y)-l-H.+iy)(a)x
X 4(.-;y) (a) sin (; v - i ; y)]
- ~ in cos (~ :re v) csch (~ :re y) X
J.(a sinh x) + J_.(a sinh x) X [L!(.+i Y) (~ a) I! (.-iy) (~ a)-
-LH.-iY)(~ a)IH.+iY)(~ a)]
~ i:re sin (~ :rev) csch (~ :re y) X
E. (a sinh x) -E_. (a sinh x) X [L!(.-i Y) (~ a) IH.+iy) (~ a)-
- L~(.+iY) (~ a) I! (.-iy) (~ a)]
ctn (~ :rex) [Jx(a)-J_x(a)] sin (a sin y) O<y<:re 0 y>:re
2 sin (a cos y) 1
csc(~ :rex) [Jx(a)-J_x(a)] O<y<-:re
2
2
Jx(a) +J_x(a) cos (a sin y) O<y<:re 0 y> :re
sec (~ :rex) [Jx (a)+J_ x (a)] 2 cos (a cos y) O<y<~:re
2
+ l-;x(a) -J_tx(a)]
XV [Hv(a x) - Y,:(ax)]
arc cos ( ~) 0
arc sin y
O<y<a
O<y<a
- 2 n-1 (y2 - a2) -t arc ctn [a-1 (y2 - a2)~]
y> a
0 y>1
X:A(~ -1',1;~ -1';1- y2 a-2)
C r 2v n-1 2" + v cos (nv) av- 1 r(1 + v) X
X y-2. J\( ~ ,1; ~ + v; 1 - y2 a-2)
(1 tt = 2" n cos (nv) r(1 + 2'1') X
X aV y-V-t (a2 - y2)-l(vH ) \l5:::::::t(;)
2n-1(a2+ y2)-k log [ a+(y:+a2)l]
log y+(a2+y2)i
x-'[l,(ax) -L,(ax)J n-lr'a'+l[r(~ +JI)t y-1X
X J\ ( 1, 1; ~ + v; - a2 y-2)
x-' [L,(a x) -L,(ax)J
X{(V+ ~ra2y-$'-1J\(~ +v, ~ +v; Rev<~ 2
~ + V; - a2 y-2) + nal- 2v tan (nv) }
x'+l[I.(ax) -L.(ax)] 2'+1 a,-l [r( - V)J-1 y-2v-1 X
-1 <Rev <1 X 21';. (1, ~; -v; - y2 a-2)
Ho (a X2) ~ na-1y{[li(;;)r-[Y-d;;W}
xl H-1 (a2 X2) 1 - eye ) - 2: a 2 2:ny Y-i "4y2a-2
xi H-i (a2 X2) - ! a-4 (~ ny yi Yt (! y2 a-2)
x--1l-1 [1_. (a x) - L. (a x) J ni-r,-l r(-v) [r(i-V) r( ~ +v)r a-'x
Rev<O X(a2+y2)"J\(_v,~; i-v; __ /22) 2 a +y
10(ax2) - Lo (a X2) ri(2nat1y [K1(;;)r
x-1 [Ho (a X-1) - Yo (a x-1) J 4n-1 Ko [(2ia y)iJ Ko [( - 2ia y)lJ
x-1 [10 (a x-1) - Lo(ax-1)J 210 [(2a y)lJ Ko [(2a y)lJ
(a2 + x2)-lx log a[(b2+y2)l_yJ}x x{10[b(a2+ x2)lJ-
- Lo [b (a2 + x2)lJ) XKog a [(b2+ y2)!+ yJ}
(a2 + X2) -l Ho [b (a2 + x2)lJ 0 y>b
§ 28. Elliptische IntegraJe 105
10 (2a cosh ~ x)- - Lo ( 2 a cosh ~ x)
g(y) =fl(x)cos(xy)dx o
XKog a [y - (y2 - b2)~]}
§ 28. Elliptische Integrate
g(y) =fl(x)cos(xy)dx o
- : n 210 (~ a y) Yo (~ a y)
1 • _1_ [1. ( 1 ) ( n 1 ) - 4 n 2 y 2 0 2 a y cos 4 + 2 a y +
+ Yo (~ a y) sin (; + ~ a y) ]
~ nKo{~ y[a + (a2- b2)~]} X
X log y[a - (a2- b2)t]} a> b
106 r. FOURIER-Kosinus-Transformationen
t(x) g(y) =Jt(x)cos(xy)dx
0 x>a
x>a
0 O<x<1
(1 + x)-!K[(;~~ /] x>1
o O<x<a
sech (a x) K[tanh(ax)]
x K [( a cosh x - 1 )~] a cosh ",+1
o
: n2[Yo(~ ay)r
a>b
~ n2{[YoC aylf - [Jo(~ ay)n
+ n2 { Yo [~ y (a + b) 1 Yo [~ y (a - b) 1 -
-Jo[~ y(a+b)llo[~ y(a-b)]}
+ P- Hiy [- (1 - a-2)lJ}
§ 29. Parabolische Zylinderfunktionen 107
e!x'D_2(x) (~ nyelY' D_ 2(y)
n = 0, 1, 2, ...
et a' x' Dv (a x) • n!2H!v[r(- ~ v)r1a!(I+V) y-Hv+3) X
Rev <0 " a-, ( y2 ) X e"Y 111 (v-l),1(v+l) 2a2
e-ia1x' D. (a x) 2HV-l)n![r(1 - ~ v)r1a-1 X
( . 1. y2 ) X 11\ 1, 1 - - v, - -2 2 2a
2Ho-I'-I)n!r(1 +p) [r(1 +±p-±v)r1x
xl' e-ix' Do (x) C 1 1 Rep> -1
X~ 2+2 P,1+ 2 P;
1 1 1 1 2) 2,1+ 2 P- 2 V;-2 Y
D20-! [(2 x)!] X n!sin (: - vn) y-2°-!(1 + y2)-! X X {D_ 2v - I [(2X)!] + + D_ 20- l [- (2x)l]} X [1 + (1 + y2)lJ2.
e-!X'[D2o _ l (x) + 2!-2v n~ sin (~ + v n) y2V-t e-ty'
+ D20-1,(- x)] 4 ,
-v -la'r'D ( -~) x e 2v-l a x (~ n)~2vy.-le-aY~Sin(~ vn-a yt)
x-v eta' x D (a xl) (;Y[y + (a + y!)2]V-! X 20-1
X cos [(2V - 1) arc tan (a;lyl) - : 1 Rev <1
x-·-1 eia' x D (a x!) - (~ nyv-l[y + (a + yl)2]' X
20-1
g(yl = j/(xlcos(xyldx o
xr(-~ v-i ~ Y)Jf!('+1l,i!y(2a2)
Rev > -~, Re(v + k)<O 2
xPWk,.(ax) W_k,.(ax)
x2• e-1x' M. (~X2) k,' 2
1 . 1 - - <Rev<-- +Rek
Re(k -v) <~ 2
-20-1 -lx' W, ( 1 2) X e k,' "2 x
Rev<~ 4
n!22v+2k a2k r(1 + 2 v) [r(1 - v - k)]-1 X
X y-2v-2k-1 3.F; (~ - k, 1 - k,
~-k+vo1-2k -V-k o_ y2a-2) 2 ' J J
~ al - 2p F(,u + v) F(,u - v) F(2,u) X 2
x[F(~ +k+,u)r(~ -k+,u)r1x X 41'; 0, ~ +,u,,u + v, ,u - v;
~ + k +,u, ~ - k +,u, ~ ; - y2 a-2)
n! 210-!k r(1 + 2v) [r(k - V)]-1 X
k-.-l -h'M ( 1 2) X Y e H1+k+ 3 .l,!{k-o-ll "2 y
§ 31. Thetafunktionen 109
1 Rev <"4' Re(k -v) <0
X21'-l e-!x'W, (X2) k,.
Rek> -~ Rev> - ~ 4' 2
XM'-I',-k-l( ~ X2)
nl2Hk-a.-l) r( ~ - 2V) [re + v-k) r1 X
X v-k-l ty'w, (1 2) y e~ Hk+av),Hk-v) 2" y
~ r(~ +V+#)r(~ -V+#)X
X [r(1 - k + #)]-1 ~2 (~ + v + fl, 1 1 1 ) 2" - v + #; 2"' 1 - k + #; - 4 y2
~ (~ nyr(1 +2v) [r(~ + k)r1x
xMHk-V)'lk-l(~ y2)
XW.+k'I'-l(~ y2)M'_k'_I'_Jl(~ y2)
X Du [(2a)!cosh (~ Y) 1
n2k (~ aY'r(1- 2k) X
X ela [sinh (ly)]' D 2k- 1 [(2a)! cosh (~ y) 1
§ 31. Thetafunktionen
g(y) = il(x) cos (x y) d x o
~ (2Hiy - 1) (1 - 2l - iy) n-1- iiy X 2
xr(: + ~ iY)C(~ +iy)
1 -:n; 2
Si (a y)
(2:n;)ly-l[ ~ - S(a y)]
(2:n; y)!
sin (a y) Ci (a y) - cos (a y) si (a y)
sin (ay) Ci [y(a + b)] - cos (ay) si [y(a+ b)]
0
n=1,2,3,···
g(y) = f/(x) sin (x y) d x o
n-l
L i:=:1!! cos [~ (n - m) - bY] x m=l
( )n-l x(a+b)-m(_y)n-m-l_ -y X (n-i)! X [cos (ay + ~ n) Ci(ay + by) +
+ sin (ay + ~ n) si(ay + by)]
(2:fi +na~sin (ay) [1-C(ay)-S (ay)]-
- na!cos(ay) [C(ay) - S(ay)J
a-I [cos (ay) si(ay + by) -
- sin (ay) Ci(ay + by) - si(by)]
na-i{cos(ay) [C(ay) - S(ay)]--
(2 n)t y-! {[ ~ - S (a y) ] cos (a y) -
- [: - C(ay)] sin (ay)}
sin (a y) Ci (a y) - cos (a y) [~ n + Si (a y) ]
ni (2y)-! [cos (ay) - sin (ay) +
+ 2C(ay + by) sin (ay) -
- 2S(ay + by) cos (ay)]
ni(2y)-i[sin(ay) + cos (ay)J
112 II. FOURIER-Sinus-Transformationen
t(x) g(y) ~'!t(x)sin(xy)dx o
0 O<x<a n~y-!sin(: + ay)-na![C(ay) -S(ay)J X-I (X - a)~ x>a
0 O<x<a na-![C(ay) - S(ay)J X-I (X - a)-~ X> a
0 O<x<a n(2a)-~{[C(2ay) - S(2ay)J cos (ay) - (x - a)-!(a + X)-I x> a - [1 - C(2ay) - S(2ay)J sin (ay)}
x-lea + x + (2ax)!J-I na-I(2a)-! {2(a;y_
- 1 + eay Erfc[(aY)!J}
0 O<xb X cos (ay) - [1 - C(ay + by)-
- S(ay + by)J sin (ay)}
~ n Erfc [(i a y)!J Erfc [( - i a y)~J 2
X-I (a + x)-II [(X + a)~ - aliJ = ~ n{1 + 2C(ay) [C(ay) -1J + + 2S(ay) [S(ay) -1J}
(a2+ X2)-1 (2a)-1 [e-aYRi(ay) - eaYEi(- ay)J
x(a2+ X2)-1 ~ne-ay 2
- [b2+ (a + X)2J-I}
-(a-x) [b2+ (a-x)2J-I
§ 1. Algebraische Funktionen 113
0
1 - n a-2(1 - cos ay) 2
X-I (a2 - X2)-1 Das Integral ist als CAUCHY-Hauptwert definiert.
e-nyY a-I So ), (a y) -..!.. na-~cos(ay) 2 ,- 2
x-! (a2 - X2)-1 Das Integral ist als CAUCHY-Hauptwert definiert.
X-I (a2 + X2)-1 1 -2(1 -a y ) -na -e 2
(a2 + x2)-i 1 -n[Io(ay) -Lo(ay)J 2
x(a2+x2)-~-1 1 aK1(ay)-- y
[X + (a2+ X2)!]-1 ..!.. n(ay)-1 [II (ay) -Ll(ay)] 2
(a + x)-I (2ny)! Hi - 2S CVar)] sin (ay) +
+ [1-2C(Vay)]cos(ay)} -
._--
0 O<x<a ..!..nlo(ay)
(X2 - a2)-! x>a 2
0 O<x<a - ..!.. n2y [Ho (a y) Yl(ay) + X-l(X2_ a2)-!
4 x>a + Yo(ay) H_l(ay)]
Oberhettinger, Tabellen 8
114 I

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