Integral transforms
A function g(y) of a variable y (which may be complex) is called the integral transform of a function f(x) with respect to a kernel K(x, y) when $$g(y) = \int\limits_a^b {K(x,y)f(x)} dx.$$.
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Literature
5nz=tanhz cnz=dnz=sechz
+ e1 + 3e1 cosech [z(3e1f"21] •
P(z) =
K=oo
(nK)
1
q=l. e1=e2=-Tea'
1 Ii k'-2 exp -]{' = 16 k~
g2
=
3e;, ga = e:.
Literature BELLMAN, R.: A brief introduction to theta functions. New York: Holt, Rinehart and Winston 1961. BYRD, P., and M. FRIEDMAN: Handbook of elliptic integrals for engineers and physicists: BerlinjGottingenjHeidelberg: Springer 1954. ERDELYI, A.: Higher transcendental functions, Vol. 2. New York: Mc-GrawHill 1953. HANCOCK, H.: Theory of elliptic functions. New York: Dover 1958. - Elliptic integrals. New York: Wiley 1917. JAHNKE, E., F. EMDE and F. LOESCH: Tables of higher functions. New York: M:cGraw-Hill1960. MILNE-THOMSON, L. M.: Jacobian elliptic function tables. New York: Dover 1950. MACRoBERT, T. M.: Functions of a complex variable. London: MacMillan 1958. NEVILLE, E. H.: Jacobian elliptic functions. Oxford 1951. OBERHETTINGER, F., and W. MAGNUS: Anwendung der elliptischen Funktionen in Physik und Technik. BerlinjGottingenjHeidelberg: Springer 1949. SCHULER, M., und H. GEBELEIN: Five place tables of elliptic functions. BerlinjGottingenjHeidelberg: Springer 1955. TRICOMI, F.: Elliptische Funktionen. BerlinjGottingen/Heidelberg: Springer 1948. WHITTAKER, E. T., and G. N. WATSON: A course of modern analysis. Cambridge 1944.
Chapter XI
XI. Integral transforms A function g(y) of a variable y (which may be complex) is called the integral transform of a function t (x) with respect to a kernel K (x, y) when g(y) =
f
a
b
K(x. y) t(x) dx.
W. Magnus et al., Formulas and Theorems for the Special Functions of Mathematical Physics © Springer-Verlag Berlin Heidelberg 1966
396
XI. Integral transforms
The kernel functions K (x, y) and the limits a, b of the interval considered here are the following most frequently encountered in applications. a
0
I
b
1
00
---0
00
(n-2 f cos (xy)
Fourier cosine
1
Fourier sine
2
(2 ;Tt) -"2 eixy
exponential Fourier
3
e- Xy
One sided Laplace
4
e- XY
Two sided Laplace
5
x y- 1
Mellin
6
2 I
(!? sin
----
00
00
00
00
---0 00 --
(xy)
I
---00 0 --
Type of transform
K(x. y)
I
0
00
(xy)"2 J~ (xy)
Hankel
00 0 - -- -
1
00
Kix(Y)
Lebedev
\l! -..!..+jx(y)
Mehler
\l!~..!.. +ix (y)
generalised Mehler
-10
e-h(x-y)'
Gauss (or Weierstrass)
11
2
- -- 1
00
-=-:~I
7
-8 -9
--
2
These transform types listed before admit (under certain conditions) explicit inversion formulas: 1. Fourier cosine transform 1
i
g(y} =
(!)"2
t(x} =
2 ..!.. (n-)2 !
00
t(x) cos (xy) dx, g(y) cos (xy) dy.
XI. Integral transforms 2. Fourier sine transform
(2).!.. j
00
I(x) sin (xy) dx,
(2).!.. j
00
g(y) sin (xy) dy.
g(y) = -;; I(x) = -;;
2
2
3. Exponential Fourier transform 1
2n/
1\g(y) = (
f
I(x) i
j
g(y)
00
1Pl
dx,
-00
nr
1
I(x) = (21 2
-00
e-''"'' dy.
4. One sided Laplace transformation
g(y) =
00
f
o
I(x) e-'"" dx,
1
cHoo
I(x) = -nz 2. f . g(y) e'"" dy. C-tOO 5. Two sided Laplace transform 00
f
g(y) =
I(x) e-'"" dx,
-00
1
I(x) = -n~ 2.
c+.oo
f.
g(y) e'
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