Fourier Transform of Gaussian over Gaussian (sort of)
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I need help with the inverse fourier transform of the following expression
$$frac{ae^{-w^2/2}}{ae^{-w^2/2}+b}$$
where $a > 0,b > 0$ and $w$ is the angular frequency.
Top term is simply a Gaussian. It looks very simple except the additive term in denominator which complicates things.
fourier-transform
asked Nov 22 at 18:34
Cowboy Trader
8712
add a comment |
up vote
0
down vote
favorite
I need help with the inverse fourier transform of the following expression
$$frac{ae^{-w^2/2}}{ae^{-w^2/2}+b}$$
where $a > 0,b > 0$ and $w$ is the angular frequency.
Top term is simply a Gaussian. It looks very simple except the additive term in denominator which complicates things.
fourier-transform
asked Nov 22 at 18:34
Cowboy Trader
8712
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I need help with the inverse fourier transform of the following expression
$$frac{ae^{-w^2/2}}{ae^{-w^2/2}+b}$$
where $a > 0,b > 0$ and $w$ is the angular frequency.
Top term is simply a Gaussian. It looks very simple except the additive term in denominator which complicates things.
fourier-transform
asked Nov 22 at 18:34
Cowboy Trader
8712
I need help with the inverse fourier transform of the following expression
$$frac{ae^{-w^2/2}}{ae^{-w^2/2}+b}$$
where $a > 0,b > 0$ and $w$ is the angular frequency.
Top term is simply a Gaussian. It looks very simple except the additive term in denominator which complicates things.
fourier-transform
fourier-transform
asked Nov 22 at 18:34
Cowboy Trader
8712
asked Nov 22 at 18:34
Cowboy Trader
8712
asked Nov 22 at 18:34
Cowboy Trader
8712
asked Nov 22 at 18:34
Cowboy Trader
8712
asked Nov 22 at 18:34
Cowboy Trader
8712
8712
add a comment |
add a comment |
1 Answer
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At least for $|a| < |b|$, you can express your function as a convergent series in powers of $a$:
$$ sum_{n=1}^infty (-1)^{n+1} frac{a^n}{b^n} e^{-nw^2/2}$$
and then transform term-by-term.
answered Nov 22 at 19:12
Robert Israel
317k23206457
What is this series called? Can it be applied to any function with af(x)/(af(x) + b)?
– Cowboy Trader
Nov 23 at 6:07
It's just a geometric series.
– Robert Israel
Nov 23 at 20:19
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
At least for $|a| < |b|$, you can express your function as a convergent series in powers of $a$:
$$ sum_{n=1}^infty (-1)^{n+1} frac{a^n}{b^n} e^{-nw^2/2}$$
and then transform term-by-term.
answered Nov 22 at 19:12
Robert Israel
317k23206457
What is this series called? Can it be applied to any function with af(x)/(af(x) + b)?
– Cowboy Trader
Nov 23 at 6:07
It's just a geometric series.
– Robert Israel
Nov 23 at 20:19
add a comment |
up vote
2
down vote
At least for $|a| < |b|$, you can express your function as a convergent series in powers of $a$:
$$ sum_{n=1}^infty (-1)^{n+1} frac{a^n}{b^n} e^{-nw^2/2}$$
and then transform term-by-term.
answered Nov 22 at 19:12
Robert Israel
317k23206457
What is this series called? Can it be applied to any function with af(x)/(af(x) + b)?
– Cowboy Trader
Nov 23 at 6:07
It's just a geometric series.
– Robert Israel
Nov 23 at 20:19
add a comment |
up vote
2
down vote
up vote
2
down vote
At least for $|a| < |b|$, you can express your function as a convergent series in powers of $a$:
$$ sum_{n=1}^infty (-1)^{n+1} frac{a^n}{b^n} e^{-nw^2/2}$$
and then transform term-by-term.
answered Nov 22 at 19:12
Robert Israel
317k23206457
At least for $|a| < |b|$, you can express your function as a convergent series in powers of $a$:
$$ sum_{n=1}^infty (-1)^{n+1} frac{a^n}{b^n} e^{-nw^2/2}$$
and then transform term-by-term.
answered Nov 22 at 19:12
Robert Israel
317k23206457
answered Nov 22 at 19:12
Robert Israel
317k23206457
answered Nov 22 at 19:12
Robert Israel
317k23206457
answered Nov 22 at 19:12
Robert Israel
317k23206457
317k23206457
What is this series called? Can it be applied to any function with af(x)/(af(x) + b)?
– Cowboy Trader
Nov 23 at 6:07
It's just a geometric series.
– Robert Israel
Nov 23 at 20:19
add a comment |
What is this series called? Can it be applied to any function with af(x)/(af(x) + b)?
– Cowboy Trader
Nov 23 at 6:07
It's just a geometric series.
– Robert Israel
Nov 23 at 20:19
What is this series called? Can it be applied to any function with af(x)/(af(x) + b)?
– Cowboy Trader
Nov 23 at 6:07
What is this series called? Can it be applied to any function with af(x)/(af(x) + b)?
– Cowboy Trader
Nov 23 at 6:07
It's just a geometric series.
– Robert Israel
Nov 23 at 20:19
It's just a geometric series.
– Robert Israel
Nov 23 at 20:19
add a comment |
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