Fourier Transform of Gaussian over Gaussian (sort of)
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
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
1
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
314k23206453
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 |
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
314k23206453
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
314k23206453
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
314k23206453
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
314k23206453
answered Nov 22 at 19:12
Robert Israel
314k23206453
answered Nov 22 at 19:12
Robert Israel
314k23206453
answered Nov 22 at 19:12
Robert Israel
314k23206453
314k23206453
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 |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3009485%2ffourier-transform-of-gaussian-over-gaussian-sort-of%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
