- #1
etf
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Fourier transform is defined as
$$F(jw)=\int_{-\infty}^{\infty}f(t)e^{-jwt}dt.$$
Inverse Fourier transform is defined as
$$f(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}F(jw)e^{jwt}dw.$$
Let ##f(t)=e^{-at}h(t),a>0##, where ##h(t)## is heaviside function and ##a## is real constant.
Fourier transform of this function is
$$F(jw)=\int_{0}^{\infty}f(t)e^{-jwt}dt=\int_{0}^{\infty}e^{-at}e^{-jwt}dt=\frac{1}{a+jw}$$
How can I calculate inverse Fourier transform of ##\frac{1}{a+jw}##, ##f(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{1}{a+jw}e^{jwt}dw##?
Although ##\frac{1}{a+jw}## doesn't look complicated, there is no way I can calculate this integral. Generaly, I have problems calculating inverse FT. Any suggestion?
Thanks in advance.
$$F(jw)=\int_{-\infty}^{\infty}f(t)e^{-jwt}dt.$$
Inverse Fourier transform is defined as
$$f(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}F(jw)e^{jwt}dw.$$
Let ##f(t)=e^{-at}h(t),a>0##, where ##h(t)## is heaviside function and ##a## is real constant.
Fourier transform of this function is
$$F(jw)=\int_{0}^{\infty}f(t)e^{-jwt}dt=\int_{0}^{\infty}e^{-at}e^{-jwt}dt=\frac{1}{a+jw}$$
How can I calculate inverse Fourier transform of ##\frac{1}{a+jw}##, ##f(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{1}{a+jw}e^{jwt}dw##?
Although ##\frac{1}{a+jw}## doesn't look complicated, there is no way I can calculate this integral. Generaly, I have problems calculating inverse FT. Any suggestion?
Thanks in advance.