Computation of complex valued integral
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I would like to get a simple formula for the complex valued integral
$$
frac{1}{sqrt{2pisigma_Z^2}}int_{-infty}^inftyexpleft(iux+delta d(t)vert xvertright)expleft(-frac{(x-mu_Z)^2}{2sigma_Z^2}right) dx.
$$
This formula is also allowed to contain the cdf $Phi$ of a standard normal RV. The above integral appears in an ODE I am trying to solve and I would like to avoid solving an integral in each step of my ODE solver. Hence, I would like to have a simple formula for the integral only containing functions that are available in most standard programming libraries.
If I replace $iu$ with $u$, i.e. only consider the real valued case I can simplify the above integral to
$$
-expleft((u+delta d(t))mu_Z+(u+delta d(t))^2frac{sigma_Z^2}{2}right)Phileft(frac{(u+delta d(t))sigma_Z^2+mu_Z}{sigma_Z}right)\
-expleft((u-delta d(t))mu_Z+(u-delta d(t))^2frac{sigma_Z^2}{2}right)Phileft(-frac{(u-delta d(t))sigma_Z^2 + mu_Z}{sigma_Z}right)
$$
However, in the complex case I would need to evaluate the cdf $Phi$ at a complex number which is implemented in most computer packages like MATLAB. Is there a way to get around this?
integration complex-analysis analysis normal-distribution complex-integration
asked Nov 23 at 23:43
lbf_1994
10910
|
show 3 more comments
up vote
0
down vote
favorite
I would like to get a simple formula for the complex valued integral
$$
frac{1}{sqrt{2pisigma_Z^2}}int_{-infty}^inftyexpleft(iux+delta d(t)vert xvertright)expleft(-frac{(x-mu_Z)^2}{2sigma_Z^2}right) dx.
$$
This formula is also allowed to contain the cdf $Phi$ of a standard normal RV. The above integral appears in an ODE I am trying to solve and I would like to avoid solving an integral in each step of my ODE solver. Hence, I would like to have a simple formula for the integral only containing functions that are available in most standard programming libraries.
If I replace $iu$ with $u$, i.e. only consider the real valued case I can simplify the above integral to
$$
-expleft((u+delta d(t))mu_Z+(u+delta d(t))^2frac{sigma_Z^2}{2}right)Phileft(frac{(u+delta d(t))sigma_Z^2+mu_Z}{sigma_Z}right)\
-expleft((u-delta d(t))mu_Z+(u-delta d(t))^2frac{sigma_Z^2}{2}right)Phileft(-frac{(u-delta d(t))sigma_Z^2 + mu_Z}{sigma_Z}right)
$$
However, in the complex case I would need to evaluate the cdf $Phi$ at a complex number which is implemented in most computer packages like MATLAB. Is there a way to get around this?
integration complex-analysis analysis normal-distribution complex-integration
asked Nov 23 at 23:43
lbf_1994
10910
$sigma=sigma_Z , ?$. Also what is $delta d(t)$?
– Diger
Nov 23 at 23:50
Yes, $sigma_Z=sigma$ (I've edited the question). $d(t)$ is the unknown function in the ODE I am trying to solve. You can just consider it to be a constant when computing the integral. The above integral is one summand in the right hand side of an ODE for $d(t)$.
– lbf_1994
Nov 23 at 23:56
And $delta$ is another constant?
– Diger
Nov 23 at 23:57
yes exactly, it is also a constant
– lbf_1994
Nov 23 at 23:59
And what do you expect from an answer? If you chuck the integral into Maple or Mathematica you will get an exponential and an error-function. Is that what you want?
– Diger
Nov 24 at 0:00
|
show 3 more comments
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I would like to get a simple formula for the complex valued integral
$$
frac{1}{sqrt{2pisigma_Z^2}}int_{-infty}^inftyexpleft(iux+delta d(t)vert xvertright)expleft(-frac{(x-mu_Z)^2}{2sigma_Z^2}right) dx.
$$
This formula is also allowed to contain the cdf $Phi$ of a standard normal RV. The above integral appears in an ODE I am trying to solve and I would like to avoid solving an integral in each step of my ODE solver. Hence, I would like to have a simple formula for the integral only containing functions that are available in most standard programming libraries.
If I replace $iu$ with $u$, i.e. only consider the real valued case I can simplify the above integral to
$$
-expleft((u+delta d(t))mu_Z+(u+delta d(t))^2frac{sigma_Z^2}{2}right)Phileft(frac{(u+delta d(t))sigma_Z^2+mu_Z}{sigma_Z}right)\
-expleft((u-delta d(t))mu_Z+(u-delta d(t))^2frac{sigma_Z^2}{2}right)Phileft(-frac{(u-delta d(t))sigma_Z^2 + mu_Z}{sigma_Z}right)
$$
However, in the complex case I would need to evaluate the cdf $Phi$ at a complex number which is implemented in most computer packages like MATLAB. Is there a way to get around this?
integration complex-analysis analysis normal-distribution complex-integration
asked Nov 23 at 23:43
lbf_1994
10910
I would like to get a simple formula for the complex valued integral
$$
frac{1}{sqrt{2pisigma_Z^2}}int_{-infty}^inftyexpleft(iux+delta d(t)vert xvertright)expleft(-frac{(x-mu_Z)^2}{2sigma_Z^2}right) dx.
$$
This formula is also allowed to contain the cdf $Phi$ of a standard normal RV. The above integral appears in an ODE I am trying to solve and I would like to avoid solving an integral in each step of my ODE solver. Hence, I would like to have a simple formula for the integral only containing functions that are available in most standard programming libraries.
If I replace $iu$ with $u$, i.e. only consider the real valued case I can simplify the above integral to
$$
-expleft((u+delta d(t))mu_Z+(u+delta d(t))^2frac{sigma_Z^2}{2}right)Phileft(frac{(u+delta d(t))sigma_Z^2+mu_Z}{sigma_Z}right)\
-expleft((u-delta d(t))mu_Z+(u-delta d(t))^2frac{sigma_Z^2}{2}right)Phileft(-frac{(u-delta d(t))sigma_Z^2 + mu_Z}{sigma_Z}right)
$$
However, in the complex case I would need to evaluate the cdf $Phi$ at a complex number which is implemented in most computer packages like MATLAB. Is there a way to get around this?
integration complex-analysis analysis normal-distribution complex-integration
integration complex-analysis analysis normal-distribution complex-integration
asked Nov 23 at 23:43
lbf_1994
10910
asked Nov 23 at 23:43
lbf_1994
10910
edited Nov 23 at 23:55
asked Nov 23 at 23:43
lbf_1994
10910
asked Nov 23 at 23:43
lbf_1994
10910
asked Nov 23 at 23:43
lbf_1994
10910
10910
$sigma=sigma_Z , ?$. Also what is $delta d(t)$?
– Diger
Nov 23 at 23:50
Yes, $sigma_Z=sigma$ (I've edited the question). $d(t)$ is the unknown function in the ODE I am trying to solve. You can just consider it to be a constant when computing the integral. The above integral is one summand in the right hand side of an ODE for $d(t)$.
– lbf_1994
Nov 23 at 23:56
And $delta$ is another constant?
– Diger
Nov 23 at 23:57
yes exactly, it is also a constant
– lbf_1994
Nov 23 at 23:59
And what do you expect from an answer? If you chuck the integral into Maple or Mathematica you will get an exponential and an error-function. Is that what you want?
– Diger
Nov 24 at 0:00
|
show 3 more comments
$sigma=sigma_Z , ?$. Also what is $delta d(t)$?
– Diger
Nov 23 at 23:50
Yes, $sigma_Z=sigma$ (I've edited the question). $d(t)$ is the unknown function in the ODE I am trying to solve. You can just consider it to be a constant when computing the integral. The above integral is one summand in the right hand side of an ODE for $d(t)$.
– lbf_1994
Nov 23 at 23:56
And $delta$ is another constant?
– Diger
Nov 23 at 23:57
yes exactly, it is also a constant
– lbf_1994
Nov 23 at 23:59
And what do you expect from an answer? If you chuck the integral into Maple or Mathematica you will get an exponential and an error-function. Is that what you want?
– Diger
Nov 24 at 0:00
$sigma=sigma_Z , ?$. Also what is $delta d(t)$?
– Diger
Nov 23 at 23:50
$sigma=sigma_Z , ?$. Also what is $delta d(t)$?
– Diger
Nov 23 at 23:50
Yes, $sigma_Z=sigma$ (I've edited the question). $d(t)$ is the unknown function in the ODE I am trying to solve. You can just consider it to be a constant when computing the integral. The above integral is one summand in the right hand side of an ODE for $d(t)$.
– lbf_1994
Nov 23 at 23:56
Yes, $sigma_Z=sigma$ (I've edited the question). $d(t)$ is the unknown function in the ODE I am trying to solve. You can just consider it to be a constant when computing the integral. The above integral is one summand in the right hand side of an ODE for $d(t)$.
– lbf_1994
Nov 23 at 23:56
And $delta$ is another constant?
– Diger
Nov 23 at 23:57
And $delta$ is another constant?
– Diger
Nov 23 at 23:57
yes exactly, it is also a constant
– lbf_1994
Nov 23 at 23:59
yes exactly, it is also a constant
– lbf_1994
Nov 23 at 23:59
And what do you expect from an answer? If you chuck the integral into Maple or Mathematica you will get an exponential and an error-function. Is that what you want?
– Diger
Nov 24 at 0:00
And what do you expect from an answer? If you chuck the integral into Maple or Mathematica you will get an exponential and an error-function. Is that what you want?
– Diger
Nov 24 at 0:00
|
show 3 more comments
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$sigma=sigma_Z , ?$. Also what is $delta d(t)$?
– Diger
Nov 23 at 23:50
Yes, $sigma_Z=sigma$ (I've edited the question). $d(t)$ is the unknown function in the ODE I am trying to solve. You can just consider it to be a constant when computing the integral. The above integral is one summand in the right hand side of an ODE for $d(t)$.
– lbf_1994
Nov 23 at 23:56
And $delta$ is another constant?
– Diger
Nov 23 at 23:57
yes exactly, it is also a constant
– lbf_1994
Nov 23 at 23:59
And what do you expect from an answer? If you chuck the integral into Maple or Mathematica you will get an exponential and an error-function. Is that what you want?
– Diger
Nov 24 at 0:00