- #1
PiRho31416
- 19
- 0
[Solved] Inverse Fourier Transform
If
[tex]F(\omega)=e^{-|\omega|\alpha}\,(\alpha>0)[/tex],
determine the inverse Fourier transform of [tex]F(\omega)[/tex]. The answer is [tex]\frac{2\alpha}{x^{2}+\alpha^{2}}[/tex]
Inverse Fourier Transform is defined as:
[tex]f(x)=\int_{-\infty}^{\infty}F(\omega)e^{-i\omega x}\, d\omega[/tex]
So I broke up the equation into two different integrals.
[tex] F(\omega)=e^{-|\omega|\alpha}=\int_{-\infty}^{\infty}e^{-|\omega|\alpha}e^{-i\omega x}d\omega=\int_{-\infty}^{0}e^{\omega(\alpha-ix)}+\int_{0}^{\infty}e^{-\omega(\alpha+ix)}d\omega [/tex]
[tex]\frac{e^{\omega(\alpha-ix)}}{\alpha-ix}\bigg|_{\omega=-\infty}^{\omega=0}+\frac{e^{-\omega(\alpha-ix)}}{\alpha-ix}\bigg|_{\omega=0}^{\omega=\infty}=\frac{1}{\alpha-ix}+\frac{1}{\alpha+ix}=\frac{\alpha+ix+\alpha-ix}{\alpha^{2}+x^{2}}=\frac{2\alpha}{\alpha^{2}+x^{2}}[/tex]
Homework Statement
If
[tex]F(\omega)=e^{-|\omega|\alpha}\,(\alpha>0)[/tex],
determine the inverse Fourier transform of [tex]F(\omega)[/tex]. The answer is [tex]\frac{2\alpha}{x^{2}+\alpha^{2}}[/tex]
Homework Equations
Inverse Fourier Transform is defined as:
[tex]f(x)=\int_{-\infty}^{\infty}F(\omega)e^{-i\omega x}\, d\omega[/tex]
The Attempt at a Solution
So I broke up the equation into two different integrals.
[tex] F(\omega)=e^{-|\omega|\alpha}=\int_{-\infty}^{\infty}e^{-|\omega|\alpha}e^{-i\omega x}d\omega=\int_{-\infty}^{0}e^{\omega(\alpha-ix)}+\int_{0}^{\infty}e^{-\omega(\alpha+ix)}d\omega [/tex]
[tex]\frac{e^{\omega(\alpha-ix)}}{\alpha-ix}\bigg|_{\omega=-\infty}^{\omega=0}+\frac{e^{-\omega(\alpha-ix)}}{\alpha-ix}\bigg|_{\omega=0}^{\omega=\infty}=\frac{1}{\alpha-ix}+\frac{1}{\alpha+ix}=\frac{\alpha+ix+\alpha-ix}{\alpha^{2}+x^{2}}=\frac{2\alpha}{\alpha^{2}+x^{2}}[/tex]
Last edited: