Question

if F(s) is the fourier transform of f (x)
then the fourier transform of f(ax) is ...​

Answer 1

SOLUTION

GIVEN

F(s) is the fourier transform of f (x)

TO DETERMINE

The fourier transform of f(ax)

EVALUATION

Here it is given that

F(s) is the fourier transform of f (x)

$\displaystyle \sf{F(s) = \int\limits_{- \infty}^{\infty} f(x)e^{isx} \, dx }$

Hence the required fourier transform of f(ax)

$\displaystyle \sf{ = \int\limits_{- \infty}^{\infty} f(ax)e^{isx} \, dx }$

$\displaystyle \sf{ = \int\limits_{- \infty}^{\infty} f(t)e^{ \frac{ist}{a} } \, \frac{dt}{a} }$

$\displaystyle \sf{ = \frac{1}{a} \int\limits_{- \infty}^{\infty} f(t)e^{ \frac{ist}{a} } \, dt}$

$\displaystyle \sf{ = \frac{1}{a} \int\limits_{- \infty}^{\infty} f(x)e^{ix .\frac{s}{a} } \, dx}$

$\displaystyle \sf{ = \frac{1}{a} F \bigg( \frac{s}{a} \bigg)}$

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