Mixing
Integrate $int_{-infty}^{infty} e^{itx}frac{1}{sqrt{2 pi}} e^{-x^2/2} dx?$
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This is the characteristic function of normal distribution. My instructor showed that we can evaluate this with differentiation under integral trick (to obtain and solve a differential function). He mentioned that this sort of integral can be generally solved through complex analysis but did not elaborate. Can someone show me how?
complex-analysis complex-integration
asked Feb 1 at 15:11
Daniel LiDaniel Li
787414
$endgroup$
add a comment |
$begingroup$
This is the characteristic function of normal distribution. My instructor showed that we can evaluate this with differentiation under integral trick (to obtain and solve a differential function). He mentioned that this sort of integral can be generally solved through complex analysis but did not elaborate. Can someone show me how?
complex-analysis complex-integration
asked Feb 1 at 15:11
Daniel LiDaniel Li
787414
$endgroup$
1
$begingroup$
Try combining the exponential terms and completing the square in the exponent. Should reduce to a form of the Gaussian integral.
$endgroup$
– aleden
Feb 1 at 15:16
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But the question is to use complex analysis.
$endgroup$
– Daniel Li
Feb 1 at 15:17
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This may help you math.stackexchange.com/questions/2741908/…, physics.stackexchange.com/questions/368186/…
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– caverac
Feb 1 at 15:18
$begingroup$
@DanielLi: to show that the complex and real integrals are equal, you will need Cauchy's Integral Theorem.
$endgroup$
– robjohn♦
Feb 1 at 15:19
add a comment |
$begingroup$
This is the characteristic function of normal distribution. My instructor showed that we can evaluate this with differentiation under integral trick (to obtain and solve a differential function). He mentioned that this sort of integral can be generally solved through complex analysis but did not elaborate. Can someone show me how?
complex-analysis complex-integration
asked Feb 1 at 15:11
Daniel LiDaniel Li
787414
$endgroup$
This is the characteristic function of normal distribution. My instructor showed that we can evaluate this with differentiation under integral trick (to obtain and solve a differential function). He mentioned that this sort of integral can be generally solved through complex analysis but did not elaborate. Can someone show me how?
complex-analysis complex-integration
complex-analysis complex-integration
asked Feb 1 at 15:11
Daniel LiDaniel Li
787414
asked Feb 1 at 15:11
Daniel LiDaniel Li
787414
asked Feb 1 at 15:11
Daniel LiDaniel Li
787414
asked Feb 1 at 15:11
Daniel LiDaniel Li
787414
asked Feb 1 at 15:11
Daniel LiDaniel Li
787414
787414
1
$begingroup$
Try combining the exponential terms and completing the square in the exponent. Should reduce to a form of the Gaussian integral.
$endgroup$
– aleden
Feb 1 at 15:16
$begingroup$
But the question is to use complex analysis.
$endgroup$
– Daniel Li
Feb 1 at 15:17
$begingroup$
This may help you math.stackexchange.com/questions/2741908/…, physics.stackexchange.com/questions/368186/…
$endgroup$
– caverac
Feb 1 at 15:18
$begingroup$
@DanielLi: to show that the complex and real integrals are equal, you will need Cauchy's Integral Theorem.
$endgroup$
– robjohn♦
Feb 1 at 15:19
add a comment |
1
$begingroup$
Try combining the exponential terms and completing the square in the exponent. Should reduce to a form of the Gaussian integral.
$endgroup$
– aleden
Feb 1 at 15:16
$begingroup$
But the question is to use complex analysis.
$endgroup$
– Daniel Li
Feb 1 at 15:17
$begingroup$
This may help you math.stackexchange.com/questions/2741908/…, physics.stackexchange.com/questions/368186/…
$endgroup$
– caverac
Feb 1 at 15:18
$begingroup$
@DanielLi: to show that the complex and real integrals are equal, you will need Cauchy's Integral Theorem.
$endgroup$
– robjohn♦
Feb 1 at 15:19
1
1
$begingroup$
Try combining the exponential terms and completing the square in the exponent. Should reduce to a form of the Gaussian integral.
$endgroup$
– aleden
Feb 1 at 15:16
$begingroup$
Try combining the exponential terms and completing the square in the exponent. Should reduce to a form of the Gaussian integral.
$endgroup$
– aleden
Feb 1 at 15:16
$begingroup$
But the question is to use complex analysis.
$endgroup$
– Daniel Li
Feb 1 at 15:17
$begingroup$
But the question is to use complex analysis.
$endgroup$
– Daniel Li
Feb 1 at 15:17
$begingroup$
This may help you math.stackexchange.com/questions/2741908/…, physics.stackexchange.com/questions/368186/…
$endgroup$
– caverac
Feb 1 at 15:18
$begingroup$
This may help you math.stackexchange.com/questions/2741908/…, physics.stackexchange.com/questions/368186/…
$endgroup$
– caverac
Feb 1 at 15:18
$begingroup$
@DanielLi: to show that the complex and real integrals are equal, you will need Cauchy's Integral Theorem.
$endgroup$
– robjohn♦
Feb 1 at 15:19
$begingroup$
@DanielLi: to show that the complex and real integrals are equal, you will need Cauchy's Integral Theorem.
$endgroup$
– robjohn♦
Feb 1 at 15:19
add a comment |
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1
$begingroup$
Try combining the exponential terms and completing the square in the exponent. Should reduce to a form of the Gaussian integral.
$endgroup$
– aleden
Feb 1 at 15:16
$begingroup$
But the question is to use complex analysis.
$endgroup$
– Daniel Li
Feb 1 at 15:17
$begingroup$
This may help you math.stackexchange.com/questions/2741908/…, physics.stackexchange.com/questions/368186/…
$endgroup$
– caverac
Feb 1 at 15:18
$begingroup$
@DanielLi: to show that the complex and real integrals are equal, you will need Cauchy's Integral Theorem.
$endgroup$
– robjohn♦
Feb 1 at 15:19