How does the following hold?
$begingroup$
For a vector valued function $f(t)in{L_{2}}~[0,infty)$ with Laplace transform of $f(t)$ being $F(s)$, how to show that the following holds:
$dfrac{1}{2pi}int_{-infty}^{infty}|F(jomega)|^2~domega=int_{0}^{infty}|f(t)|^2~dt$.
I know this directly follows from Perseval's identity but is there any other way to prove the claim using the definition of Laplace transformation $$F(s)=int_{0}^{infty}f(t)e^{-st}dt.$$
Any hint is greatly appreciated.
norm laplace-transform fourier-transform
asked Dec 10 '18 at 23:54
jbgujgujbgujgu
9711
$endgroup$
add a comment |
$begingroup$
For a vector valued function $f(t)in{L_{2}}~[0,infty)$ with Laplace transform of $f(t)$ being $F(s)$, how to show that the following holds:
$dfrac{1}{2pi}int_{-infty}^{infty}|F(jomega)|^2~domega=int_{0}^{infty}|f(t)|^2~dt$.
I know this directly follows from Perseval's identity but is there any other way to prove the claim using the definition of Laplace transformation $$F(s)=int_{0}^{infty}f(t)e^{-st}dt.$$
Any hint is greatly appreciated.
norm laplace-transform fourier-transform
asked Dec 10 '18 at 23:54
jbgujgujbgujgu
9711
$endgroup$
add a comment |
$begingroup$
For a vector valued function $f(t)in{L_{2}}~[0,infty)$ with Laplace transform of $f(t)$ being $F(s)$, how to show that the following holds:
$dfrac{1}{2pi}int_{-infty}^{infty}|F(jomega)|^2~domega=int_{0}^{infty}|f(t)|^2~dt$.
I know this directly follows from Perseval's identity but is there any other way to prove the claim using the definition of Laplace transformation $$F(s)=int_{0}^{infty}f(t)e^{-st}dt.$$
Any hint is greatly appreciated.
norm laplace-transform fourier-transform
asked Dec 10 '18 at 23:54
jbgujgujbgujgu
9711
$endgroup$
For a vector valued function $f(t)in{L_{2}}~[0,infty)$ with Laplace transform of $f(t)$ being $F(s)$, how to show that the following holds:
$dfrac{1}{2pi}int_{-infty}^{infty}|F(jomega)|^2~domega=int_{0}^{infty}|f(t)|^2~dt$.
I know this directly follows from Perseval's identity but is there any other way to prove the claim using the definition of Laplace transformation $$F(s)=int_{0}^{infty}f(t)e^{-st}dt.$$
Any hint is greatly appreciated.
norm laplace-transform fourier-transform
norm laplace-transform fourier-transform
asked Dec 10 '18 at 23:54
jbgujgujbgujgu
9711
asked Dec 10 '18 at 23:54
jbgujgujbgujgu
9711
asked Dec 10 '18 at 23:54
jbgujgujbgujgu
9711
asked Dec 10 '18 at 23:54
jbgujgujbgujgu
9711
asked Dec 10 '18 at 23:54
jbgujgujbgujgu
9711
9711
add a comment |
add a comment |
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