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Gauge Institute Journal, Volume 8, No.2, May 2012H. Vic Dannon7.δ (, xy)and the Fourier Transform7.1 2-Dimesional Delta Functionδ⎛ ωx=∞ ⎞⎛ωy=∞ ⎞1 iω1ixxωyyxy = e dω⎟⎜⎟x ⎜ e dy2π∫ 2π∫ ω⎜ ⎝ ω =−∞ ⎠⎝ ⎟⎜ω =−∞ ⎠⎟( , )xy⎛ νx=∞ ⎞⎛νy=∞ ⎞2πνi2 ixxπνyy= e dν⎟⎜⎟x ⎜ e dνy,∫∫⎜ ⎝ ν =−∞ ⎠⎝ ⎟⎜ν =−∞ ⎠⎟xyωωxy==2πν2πνxy7.2 2-Dimesional Fourier TransformF{ }y=∞ x=∞−iωx−iωyx yf (, x y) f (, x y) e dxdy= ∫ ∫y=−∞x=−∞y=∞ x=∞− 2 πi( ν x+ν y)x y= ∫ ∫ f (, x y) e dxdy ,y=−∞x=−∞ωωxy==2πν2πνxy7.3 2-Dimesional Inverse Fourier Transform−1ωy=∞ ωx=∞1i( ωxx+ωyy)x y 2x y x y(2 π)ω =−∞ ω =−∞{ } = ∫ ∫F F( ω , ω ) F( ω , ω ) e dω d ωyx26
Gauge Institute Journal, Volume 8, No.2, May 2012H. Vic Dannonνyy=∞ ν =∞= ∫ ∫ν =−∞ ν =−∞xx2 πi( ν x+ν y)x y xx yF(2 πν , 2 πν ) e dν dνy,ωωxy==2πν2πνxy7.4 2-Dimesional Fourier Integral Theoremωy=∞ ωx=∞ ⎛ η=∞ ξ=∞⎞1−iωξ x− iωη y i( ωxx+ωyy)fxy (, ) = f(, ξη)e dd ξ η e dω 2 ∫ ∫ xdω(2 π)∫ ∫ω =−∞ ω =−∞⎜⎝η=−∞ ξ=−∞⎠⎟yxη=∞ ξ=∞ ⎛ ω1 x =∞⎞ ⎛ ωy=∞⎞iω( ) 1iy( y )(, )xx−ξω −η= ∫ ∫ f ξη e dω 2xdξ e dω 2yηπ∫ π∫ η=−∞ ξ=−∞ ⎜ d⎝ ω =−∞ ⎠⎟ ⎜⎝ ω =−∞⎠⎟xη=∞ ξ=∞ ⎛ ωx=∞⎞ ⎛ ωy=∞⎞2 πν i ( )2 iy( y )(, )xx−ξπν −η= f ξη e dν xdξ e dν ydηη=−∞ ξ=−∞ ⎜ ⎝ ω =−∞ ⎠⎟ ⎜⎝ω=−∞⎠⎟∫ ∫ ∫ ∫ ,xyyyωωxy==2πν2πνxy27
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