FCCR Table of Contents
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-1Analytic Functions, Fourier and Hilbert Transforms
and Dispersion Relations for Causal Functions-0
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This appendix collects some standard material relating
the named topics.
Some general references for its contents and further
material are
[Roos 1969]
[Bremermann 1965].
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For a closed Jordan curve Γ in the complex plane and a function F(z)
analytic within Γ and continuous on Γ:
for λ within Γ
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1 %10{%00 F(z)
F(%10l%00) = %10--- |B -------%00 dz
2%10p%00i %10}%005S1%10G%00T4 z - %10l%00
for %10l%00 outside of %10G%00
1 %10{%00 F(z)
%10--- |B -------%00 dz = 0
2%10p%00i %10}%005S1%10G%00T4 z - %10l%00
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If F(z) is known and analytic on Γ, then the Cauchy integral
provides its analytic continuation to the interior of Γ.
If F(z) is not analytic on Γ, the function defined on the contour
is the boundary value of a function analytic within Γ, so the
analytic function can be used to represent the boundary value
function. As an analytic function, F(z) analytic within Γ can
have a singularity on Γ.
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For λ on Γ, assume that F(z) is continuous on Γ,
then by indenting the contour Γ at λ by a small semicircle,
one proves in the usual textbook fashion that
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1 %10{%00 F(z)
F(%10l%00) = %10--- |B -------%00 dz
%10p%00i %10}%005S1ΓT4 z - %10l%00
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where the integral in this case is understood as a Cauchy Principal
Value, if F(z) is not continuous on Γ at %10l%00
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Consider now a Γ that is semicircular, with center at the
imaginary point ia and with radius R, and let F(z) be analytic
in the upper half\-plane region Im(z) > a, and continuous in
the closed lower half\-plane Im(z) %10T%00 a. Assume further
that F(z) vanishes as z%10>%00%10I%00 in the open upper half\-plane.
Then by letting R%10>%00%10I%00 in the contour one obtains the integral
representation for %10l%00 in the upper half\-plane:
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5S1+%10I%00+iaT4
1 %10{%00 F(z)
F(%10l%00) = %10--- | -------%00 dz
2%10p%00i %10}%00 z - %10l%00
5S0-%10I%00+iaT4
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For Im(%10l%00) < a, the integral vanishes.
If Im(%10l%00) = a, again indent the contour into
the region of analyticity by a semicircle, integrate and
take the Cauchy principal value as the radius
of the semicircle approaches zero.
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Let F(%10w%00) and f(t) be two functions related
by a Fourier transform and its inverse:
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5S1+%10I%00T4
F(%10w%00) = (1/%10/%002%10p%00) %10&%00 f(t)e5S0i%10w%00tT4 dt
5S0-%10I%00T4
5S1+%10I%00T4
f(t) = (1/%10/%002%10p%00) %10&%00 F(%10w%00)e5S0-i%10w%00tT4 dw
5S0-%10I%00T4
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Define the positive and negative frequency parts of F(%10w%00)
in terms of unilateral Fourier transforms.
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5S1+%10I%00T4
F5S3+T04(%10w%00) = (1/%10/%002%10p%00) %10&%00 f(t)e5S0i%10w%00tT4 dt
5S00T4
5S00T4
F5S3-T04(%10w%00) = (1/%10/%002%10p%00) %10&%00 f(t)e5S0i%10w%00tT4 dt
5S1-%10I%00T4
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so F5S1+T4(%10w%00) and F5S1-T4(%10w%00) are
defined on the real %10w%00 axis. If f(t) is absolutely integrable
over the real axis, both F5S1+T4(%10w%00) and
F5S1-T4(%10w%00) can be analytically continued into the
upper and lower half\-planes respectively. Absolute integrability is
a sufficient but not necessary condition for analytic continuability:
if f(t) is absolutely integrable on some finite interval and at most
of exponential growth at infinity, analytic continuation is also possible.
U0
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Created: September 19, 2005