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Some profitable 'fourier' transform

NON-parametric direct expansion(s)

t 2 ^ - exp t ω fourier maple t 4 ^ t 2 ^ - exp * t ω fourier maple 3 1 mat π sqrt ω 2 ^ 4 / - exp * π sqrt ω 2 ^ 4 / - exp * 12 1 ω 2 ^ - ( * ω 4 ^ + ( * 16 / 3 1 mat

an expansion that isolates 'Heaviside' wrt 'ω'

1 1 t + / t ω fourier maple i π * i ω * exp * 1 ω - sign + 2 Heavisideω * - ( * 2 /

Mathcad 8 Pro. ... 11 <= Smath/Maple does not expand for symbolic export

t a - ( 2 ^ - to / exp t ω fourier maple a 2 ^ - to / exp π to * sqrt * 1 4 / ω 2 ^ - 4 a 2 ^ * to 2 ^ / + ( * exp * i a * ω * - exp * ≡

pre-assign for numerical evaluation wrt 'ω'

a 10 : To 1 : 1 2 mat

a Gaussian shifted by 'a'

a To t q t a - ( 2 ^ - To / exp :

'fourier wavelet' wraps the space 'x'domain => 't'domain

ω f a To t q t ω fourier maple : π sqrt 100 - exp * 10 i * ω * - exp * 400 ω 2 ^ - 4 / exp * a To x q a 2 a 1 - 2 2 mat 2 1 sys x f Im x f abs 2 1 sys

pre-assign for numerical evaluation wrt 'ω'

a 3 : To 1.5 : 1 2 mat

a Gaussian shifted by 'a'

a To t q t a - ( 2 ^ - To / exp :

'fourier wavelet' wraps the space 'x'domain => 't'domain

ω f a To t q t ω fourier maple : π sqrt 3 sqrt * 6 - exp * 3 i * ω * - exp * 3 16 ω 2 ^ - ( * 8 / exp * 2 sqrt / a To x q a 2 a 1 - 2 2 mat 2 1 sys x f Re x f Im x f abs 3 1 sys π sqrt ω 2 ^ 4 / - exp * i ω * π sqrt * ω 2 ^ 4 / - exp * 2 / - π sqrt ω 2 ^ 4 / - exp * 2 ω 2 ^ - ( * 4 / i ω * π sqrt * ω 2 ^ 4 / - exp * 6 ω 2 ^ - ( * 8 / - π sqrt ω 2 ^ 4 / - exp * 12 1 ω 2 ^ - ( * ω 4 ^ + ( * 16 / 5 1 mat

pulse train

a To x Φ a 1 - ( k ^ x k To * - sign 2 / * ( k 0 7 sum * :

experiment these two

t f # ≡ π sqrt ω 2 ^ 4 / - exp * 12 1 ω 2 ^ - ( * ω 4 ^ + ( * 16 / x f x Fo Re x Fo Im 3 1 sys 2.5 0.5 - x Φ 2.5 1 x Φ 2 1 sys