Evaluate this complex integral The 2019 Stack Overflow Developer Survey Results Are InThe distribution of the inner product of a random complex normal vector.How do I evaluate the following integral $int_-infty^infty e^-sigma^2 x^2/2; mathrm dx$?Translated complex gaussian-type integral: $int_0^infty exp(i(t-alpha)^2) dt$Gaussian integral for a vector and a function - how to evaluateObtaining cdf from a pdf containing absolute value and sign functionIntegrating the product of two Laguerre polynomials using their generating function?solve the integral of gaussian random vectorCalculating mean and covariance of a truncated multivariate GaussianIntegral involving Laguerre polynomialsApplication of an exponential whose power is a second derivative
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Evaluate this complex integral
The 2019 Stack Overflow Developer Survey Results Are InThe distribution of the inner product of a random complex normal vector.How do I evaluate the following integral $int_-infty^infty e^-sigma^2 x^2/2; mathrm dx$?Translated complex gaussian-type integral: $int_0^infty exp(i(t-alpha)^2) dt$Gaussian integral for a vector and a function - how to evaluateObtaining cdf from a pdf containing absolute value and sign functionIntegrating the product of two Laguerre polynomials using their generating function?solve the integral of gaussian random vectorCalculating mean and covariance of a truncated multivariate GaussianIntegral involving Laguerre polynomialsApplication of an exponential whose power is a second derivative
$begingroup$
I have the following complex integral that corresponds to a complex integral of a Wigner function of a 1-mode Gaussian state:
$$I_n(sigma) = int^infty_-infty d^2alpha ; L_n left(frac42 - sigmaright) expleftfrac^2sigma - 2right expleft-frac12(alpha - lambda)^top V^-1(alpha - lambda)right,$$
where $alpha$ is a complex variable, $d^2alpha = dRe(alpha)dIm(alpha)$, $lambda$ and $V$ are the first (mean) and second (covariance) moments of the Gaussian state defined through:
$$lambda = (a, b)^top, quad V = beginpmatrix
c & d \
d & e
endpmatrix,$$
where the coefficients $a,b,c,d,e$ are general complex numbers, and $L_n(x)$ are the Laguerre polynomials.
I am aware that this type of integral is usually evaluated by using polar coordinates. However, I am unsure how to achieve this.
Any help is appreciated. Even a method that obtains the solution numerically is fine.
calculus integration probability-distributions mathematical-physics complex-integration
edited 13 hours ago
Andrews
1,2812423
asked Apr 6 at 17:44
SidSid
23219
$endgroup$
add a comment |
$begingroup$
I have the following complex integral that corresponds to a complex integral of a Wigner function of a 1-mode Gaussian state:
$$I_n(sigma) = int^infty_-infty d^2alpha ; L_n left(frac42 - sigmaright) expleftfrac^2sigma - 2right expleft-frac12(alpha - lambda)^top V^-1(alpha - lambda)right,$$
where $alpha$ is a complex variable, $d^2alpha = dRe(alpha)dIm(alpha)$, $lambda$ and $V$ are the first (mean) and second (covariance) moments of the Gaussian state defined through:
$$lambda = (a, b)^top, quad V = beginpmatrix
c & d \
d & e
endpmatrix,$$
where the coefficients $a,b,c,d,e$ are general complex numbers, and $L_n(x)$ are the Laguerre polynomials.
I am aware that this type of integral is usually evaluated by using polar coordinates. However, I am unsure how to achieve this.
Any help is appreciated. Even a method that obtains the solution numerically is fine.
calculus integration probability-distributions mathematical-physics complex-integration
edited 13 hours ago
Andrews
1,2812423
asked Apr 6 at 17:44
SidSid
23219
$endgroup$
add a comment |
$begingroup$
I have the following complex integral that corresponds to a complex integral of a Wigner function of a 1-mode Gaussian state:
$$I_n(sigma) = int^infty_-infty d^2alpha ; L_n left(frac42 - sigmaright) expleftfrac^2sigma - 2right expleft-frac12(alpha - lambda)^top V^-1(alpha - lambda)right,$$
where $alpha$ is a complex variable, $d^2alpha = dRe(alpha)dIm(alpha)$, $lambda$ and $V$ are the first (mean) and second (covariance) moments of the Gaussian state defined through:
$$lambda = (a, b)^top, quad V = beginpmatrix
c & d \
d & e
endpmatrix,$$
where the coefficients $a,b,c,d,e$ are general complex numbers, and $L_n(x)$ are the Laguerre polynomials.
I am aware that this type of integral is usually evaluated by using polar coordinates. However, I am unsure how to achieve this.
Any help is appreciated. Even a method that obtains the solution numerically is fine.
calculus integration probability-distributions mathematical-physics complex-integration
edited 13 hours ago
Andrews
1,2812423
asked Apr 6 at 17:44
SidSid
23219
$endgroup$
I have the following complex integral that corresponds to a complex integral of a Wigner function of a 1-mode Gaussian state:
$$I_n(sigma) = int^infty_-infty d^2alpha ; L_n left(frac42 - sigmaright) expleftfrac^2sigma - 2right expleft-frac12(alpha - lambda)^top V^-1(alpha - lambda)right,$$
where $alpha$ is a complex variable, $d^2alpha = dRe(alpha)dIm(alpha)$, $lambda$ and $V$ are the first (mean) and second (covariance) moments of the Gaussian state defined through:
$$lambda = (a, b)^top, quad V = beginpmatrix
c & d \
d & e
endpmatrix,$$
where the coefficients $a,b,c,d,e$ are general complex numbers, and $L_n(x)$ are the Laguerre polynomials.
I am aware that this type of integral is usually evaluated by using polar coordinates. However, I am unsure how to achieve this.
Any help is appreciated. Even a method that obtains the solution numerically is fine.
calculus integration probability-distributions mathematical-physics complex-integration
calculus integration probability-distributions mathematical-physics complex-integration
edited 13 hours ago
Andrews
1,2812423
asked Apr 6 at 17:44
SidSid
23219
edited 13 hours ago
Andrews
1,2812423
asked Apr 6 at 17:44
SidSid
23219
edited 13 hours ago
Andrews
1,2812423
edited 13 hours ago
Andrews
1,2812423
edited 13 hours ago
Andrews
1,2812423
1,2812423
asked Apr 6 at 17:44
SidSid
23219
asked Apr 6 at 17:44
SidSid
23219
asked Apr 6 at 17:44
SidSid
23219
23219
add a comment |
add a comment |
0
active
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