.nf -1Analytic Functions, Fourier and Hilbert Transforms and Dispersion Relations for Causal Functions-0 .sp 2 .fi This appendix collects some standard material relating the named topics. Some general references for its contents and further material are [Roos 1969] [Bremermann 1965]. .PP .PP 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 .sp .ls 1 .nf 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 .fi .ls 2 .sp 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. .PP 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 .sp .ls 1 .nf 1 %10{%00 F(z) F(%10l%00) = %10--- |B -------%00 dz %10p%00i %10}%005S1ΓT4 z - %10l%00 .fi .ls 2 .sp where the integral in this case is understood as a Cauchy Principal Value, if F(z) is not continuous on Γ at %10l%00 .PP 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: .sp .ls 1 .nf 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 .fi .ls 2 .sp 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. .bp .sp .nf Let F(%10w%00) and f(t) be two functions related by a Fourier transform and its inverse: .ls 1 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 .fi .ls 2 .sp Define the positive and negative frequency parts of F(%10w%00) in terms of unilateral Fourier transforms. .sp .ls 1 .nf 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 .fi .ls 2 .sp 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