How to evaluate this gaussian integral
Clash Royale CLAN TAG#URR8PPP
up vote
4
down vote
favorite
Is there a way to solve this integral?
$$int_-infty^infty e^-(s^2+2sqrtts)sin(y+2sqrtts),ds$$
I usually apply this formula for integrals of this kind
$$int_-infty^infty e^-(as^2+bs+c),ds=sqrtfracpiae^fracb^24a-c$$
Thank you in advance
calculus integration
New contributor
add a comment |Â
up vote
4
down vote
favorite
Is there a way to solve this integral?
$$int_-infty^infty e^-(s^2+2sqrtts)sin(y+2sqrtts),ds$$
I usually apply this formula for integrals of this kind
$$int_-infty^infty e^-(as^2+bs+c),ds=sqrtfracpiae^fracb^24a-c$$
Thank you in advance
calculus integration
New contributor
3
Use $sin(y+2sqrtts)=frace^i(y+2sqrtts)-e^-i(y+2sqrtts)2i$ and then use your formula.
â aleden
2 hours ago
3
With complex integration, this is easy (your last formula is valid for any complex $b$ - as both sides represent entire functions of $b$ - and that's enough to get the answer right away).
â metamorphy
2 hours ago
add a comment |Â
up vote
4
down vote
favorite
up vote
4
down vote
favorite
Is there a way to solve this integral?
$$int_-infty^infty e^-(s^2+2sqrtts)sin(y+2sqrtts),ds$$
I usually apply this formula for integrals of this kind
$$int_-infty^infty e^-(as^2+bs+c),ds=sqrtfracpiae^fracb^24a-c$$
Thank you in advance
calculus integration
New contributor
Is there a way to solve this integral?
$$int_-infty^infty e^-(s^2+2sqrtts)sin(y+2sqrtts),ds$$
I usually apply this formula for integrals of this kind
$$int_-infty^infty e^-(as^2+bs+c),ds=sqrtfracpiae^fracb^24a-c$$
Thank you in advance
calculus integration
calculus integration
New contributor
New contributor
edited 1 hour ago
DarÃo A. Gutiérrez
2,52331430
2,52331430
New contributor
asked 2 hours ago
Riccardo
263
263
New contributor
New contributor
3
Use $sin(y+2sqrtts)=frace^i(y+2sqrtts)-e^-i(y+2sqrtts)2i$ and then use your formula.
â aleden
2 hours ago
3
With complex integration, this is easy (your last formula is valid for any complex $b$ - as both sides represent entire functions of $b$ - and that's enough to get the answer right away).
â metamorphy
2 hours ago
add a comment |Â
3
Use $sin(y+2sqrtts)=frace^i(y+2sqrtts)-e^-i(y+2sqrtts)2i$ and then use your formula.
â aleden
2 hours ago
3
With complex integration, this is easy (your last formula is valid for any complex $b$ - as both sides represent entire functions of $b$ - and that's enough to get the answer right away).
â metamorphy
2 hours ago
3
3
Use $sin(y+2sqrtts)=frace^i(y+2sqrtts)-e^-i(y+2sqrtts)2i$ and then use your formula.
â aleden
2 hours ago
Use $sin(y+2sqrtts)=frace^i(y+2sqrtts)-e^-i(y+2sqrtts)2i$ and then use your formula.
â aleden
2 hours ago
3
3
With complex integration, this is easy (your last formula is valid for any complex $b$ - as both sides represent entire functions of $b$ - and that's enough to get the answer right away).
â metamorphy
2 hours ago
With complex integration, this is easy (your last formula is valid for any complex $b$ - as both sides represent entire functions of $b$ - and that's enough to get the answer right away).
â metamorphy
2 hours ago
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
4
down vote
accepted
If you're OK with working with functions of a complex variable, the formula you gave still applies if $a,b,cinmathbb C$ provided that $mathrmRe[a] > 0$. It gives
beginmultline
int_-infty^infty e^-(s^2+2sqrtts)sin(y+2sqrtts),ds = mathrmImleft[int_-infty^infty e^-(s^2+2sqrtts)e^ileft(y+2sqrttsright)dsright]
\= mathrmImleft[int_-infty^infty e^-[s^2+2(1-i)sqrtts-iy]dsright] = mathrmImleft[sqrtpiexpleft(-2it +i yright)right] = sqrtpisin(y-2t).
endmultline
If you're not OK with working with complex numbers, there actually are formulas for these:
begineqnarray
int_-infty^infty e^-(as^2 + bs + c)sin(ks + m)ds &=& sqrtfracpiae^fracb^2-k^24a - csinleft(m-fracb k2aright)\
int_-infty^infty e^-(as^2 + bs + c)cos(ks + m)ds &=& sqrtfracpiae^fracb^2-k^24a - ccosleft(m-fracb k2aright)
endeqnarray
These formulas follow directly from the complex form of the integral I used before, and if you plug in the values, you'll get $sqrtpisin(y-2t)$ again.
If you want to know how to get these formulas without using complex numbers, well, that's a more involved post.
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
accepted
If you're OK with working with functions of a complex variable, the formula you gave still applies if $a,b,cinmathbb C$ provided that $mathrmRe[a] > 0$. It gives
beginmultline
int_-infty^infty e^-(s^2+2sqrtts)sin(y+2sqrtts),ds = mathrmImleft[int_-infty^infty e^-(s^2+2sqrtts)e^ileft(y+2sqrttsright)dsright]
\= mathrmImleft[int_-infty^infty e^-[s^2+2(1-i)sqrtts-iy]dsright] = mathrmImleft[sqrtpiexpleft(-2it +i yright)right] = sqrtpisin(y-2t).
endmultline
If you're not OK with working with complex numbers, there actually are formulas for these:
begineqnarray
int_-infty^infty e^-(as^2 + bs + c)sin(ks + m)ds &=& sqrtfracpiae^fracb^2-k^24a - csinleft(m-fracb k2aright)\
int_-infty^infty e^-(as^2 + bs + c)cos(ks + m)ds &=& sqrtfracpiae^fracb^2-k^24a - ccosleft(m-fracb k2aright)
endeqnarray
These formulas follow directly from the complex form of the integral I used before, and if you plug in the values, you'll get $sqrtpisin(y-2t)$ again.
If you want to know how to get these formulas without using complex numbers, well, that's a more involved post.
add a comment |Â
up vote
4
down vote
accepted
If you're OK with working with functions of a complex variable, the formula you gave still applies if $a,b,cinmathbb C$ provided that $mathrmRe[a] > 0$. It gives
beginmultline
int_-infty^infty e^-(s^2+2sqrtts)sin(y+2sqrtts),ds = mathrmImleft[int_-infty^infty e^-(s^2+2sqrtts)e^ileft(y+2sqrttsright)dsright]
\= mathrmImleft[int_-infty^infty e^-[s^2+2(1-i)sqrtts-iy]dsright] = mathrmImleft[sqrtpiexpleft(-2it +i yright)right] = sqrtpisin(y-2t).
endmultline
If you're not OK with working with complex numbers, there actually are formulas for these:
begineqnarray
int_-infty^infty e^-(as^2 + bs + c)sin(ks + m)ds &=& sqrtfracpiae^fracb^2-k^24a - csinleft(m-fracb k2aright)\
int_-infty^infty e^-(as^2 + bs + c)cos(ks + m)ds &=& sqrtfracpiae^fracb^2-k^24a - ccosleft(m-fracb k2aright)
endeqnarray
These formulas follow directly from the complex form of the integral I used before, and if you plug in the values, you'll get $sqrtpisin(y-2t)$ again.
If you want to know how to get these formulas without using complex numbers, well, that's a more involved post.
add a comment |Â
up vote
4
down vote
accepted
up vote
4
down vote
accepted
If you're OK with working with functions of a complex variable, the formula you gave still applies if $a,b,cinmathbb C$ provided that $mathrmRe[a] > 0$. It gives
beginmultline
int_-infty^infty e^-(s^2+2sqrtts)sin(y+2sqrtts),ds = mathrmImleft[int_-infty^infty e^-(s^2+2sqrtts)e^ileft(y+2sqrttsright)dsright]
\= mathrmImleft[int_-infty^infty e^-[s^2+2(1-i)sqrtts-iy]dsright] = mathrmImleft[sqrtpiexpleft(-2it +i yright)right] = sqrtpisin(y-2t).
endmultline
If you're not OK with working with complex numbers, there actually are formulas for these:
begineqnarray
int_-infty^infty e^-(as^2 + bs + c)sin(ks + m)ds &=& sqrtfracpiae^fracb^2-k^24a - csinleft(m-fracb k2aright)\
int_-infty^infty e^-(as^2 + bs + c)cos(ks + m)ds &=& sqrtfracpiae^fracb^2-k^24a - ccosleft(m-fracb k2aright)
endeqnarray
These formulas follow directly from the complex form of the integral I used before, and if you plug in the values, you'll get $sqrtpisin(y-2t)$ again.
If you want to know how to get these formulas without using complex numbers, well, that's a more involved post.
If you're OK with working with functions of a complex variable, the formula you gave still applies if $a,b,cinmathbb C$ provided that $mathrmRe[a] > 0$. It gives
beginmultline
int_-infty^infty e^-(s^2+2sqrtts)sin(y+2sqrtts),ds = mathrmImleft[int_-infty^infty e^-(s^2+2sqrtts)e^ileft(y+2sqrttsright)dsright]
\= mathrmImleft[int_-infty^infty e^-[s^2+2(1-i)sqrtts-iy]dsright] = mathrmImleft[sqrtpiexpleft(-2it +i yright)right] = sqrtpisin(y-2t).
endmultline
If you're not OK with working with complex numbers, there actually are formulas for these:
begineqnarray
int_-infty^infty e^-(as^2 + bs + c)sin(ks + m)ds &=& sqrtfracpiae^fracb^2-k^24a - csinleft(m-fracb k2aright)\
int_-infty^infty e^-(as^2 + bs + c)cos(ks + m)ds &=& sqrtfracpiae^fracb^2-k^24a - ccosleft(m-fracb k2aright)
endeqnarray
These formulas follow directly from the complex form of the integral I used before, and if you plug in the values, you'll get $sqrtpisin(y-2t)$ again.
If you want to know how to get these formulas without using complex numbers, well, that's a more involved post.
answered 59 mins ago
eyeballfrog
5,301528
5,301528
add a comment |Â
add a comment |Â
Riccardo is a new contributor. Be nice, and check out our Code of Conduct.
Riccardo is a new contributor. Be nice, and check out our Code of Conduct.
Riccardo is a new contributor. Be nice, and check out our Code of Conduct.
Riccardo is a new contributor. Be nice, and check out our Code of Conduct.
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
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2973686%2fhow-to-evaluate-this-gaussian-integral%23new-answer', 'question_page');
);
Post as a guest
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
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
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
3
Use $sin(y+2sqrtts)=frace^i(y+2sqrtts)-e^-i(y+2sqrtts)2i$ and then use your formula.
â aleden
2 hours ago
3
With complex integration, this is easy (your last formula is valid for any complex $b$ - as both sides represent entire functions of $b$ - and that's enough to get the answer right away).
â metamorphy
2 hours ago