Integral $intlimits_0^infty text{d}q cos(qx) (omega^2-q^2)^{N-1} e^{-s(omega^2-q^2)^N}$
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In my research I am interested in finding the following expression:
$$ f{}_omega^N(x) := intlimits_0^{ell^{,2N}} text{d}s ~ mathcal{I}{}_omega^N(x,s) $$
where the main problem lies in finding an analytic expression for
$$ mathcal{I}{}_omega^N(x,s) := intlimits_0^infty frac{text{d}q}{pi} cos(qx) (omega^2-q^2)^{N-1} e^{-s(omega^2-q^2)^N} , , quad N ge 2 , , quad s > 0 , , quad x in mathbb{R} , .$$
For $s>0$ this integral is convergent, and it can be evaluated numerically in principle. However, it would be nice to have an analytic expression.
In the case $N=1$ the result can be specified in terms of the imaginary error function and one obtains
$$ mathcal{I}^1_omega(x,s) = frac{e^{x^2/(4s)-s,omega^2}}{4sqrt{pi s}}left[ text{erfi}left(frac{2qs-ix}{2sqrt{s}}right) + text{erfi}left(frac{2qs+ix}{2sqrt{s}}right) right] , ,$$
but in a general case (or even just for $N=2$) I cannot solve the integral. Any ideas?
integration definite-integrals
asked Nov 15 at 19:40
Jens
316
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up vote
1
down vote
favorite
In my research I am interested in finding the following expression:
$$ f{}_omega^N(x) := intlimits_0^{ell^{,2N}} text{d}s ~ mathcal{I}{}_omega^N(x,s) $$
where the main problem lies in finding an analytic expression for
$$ mathcal{I}{}_omega^N(x,s) := intlimits_0^infty frac{text{d}q}{pi} cos(qx) (omega^2-q^2)^{N-1} e^{-s(omega^2-q^2)^N} , , quad N ge 2 , , quad s > 0 , , quad x in mathbb{R} , .$$
For $s>0$ this integral is convergent, and it can be evaluated numerically in principle. However, it would be nice to have an analytic expression.
In the case $N=1$ the result can be specified in terms of the imaginary error function and one obtains
$$ mathcal{I}^1_omega(x,s) = frac{e^{x^2/(4s)-s,omega^2}}{4sqrt{pi s}}left[ text{erfi}left(frac{2qs-ix}{2sqrt{s}}right) + text{erfi}left(frac{2qs+ix}{2sqrt{s}}right) right] , ,$$
but in a general case (or even just for $N=2$) I cannot solve the integral. Any ideas?
integration definite-integrals
asked Nov 15 at 19:40
Jens
316
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
In my research I am interested in finding the following expression:
$$ f{}_omega^N(x) := intlimits_0^{ell^{,2N}} text{d}s ~ mathcal{I}{}_omega^N(x,s) $$
where the main problem lies in finding an analytic expression for
$$ mathcal{I}{}_omega^N(x,s) := intlimits_0^infty frac{text{d}q}{pi} cos(qx) (omega^2-q^2)^{N-1} e^{-s(omega^2-q^2)^N} , , quad N ge 2 , , quad s > 0 , , quad x in mathbb{R} , .$$
For $s>0$ this integral is convergent, and it can be evaluated numerically in principle. However, it would be nice to have an analytic expression.
In the case $N=1$ the result can be specified in terms of the imaginary error function and one obtains
$$ mathcal{I}^1_omega(x,s) = frac{e^{x^2/(4s)-s,omega^2}}{4sqrt{pi s}}left[ text{erfi}left(frac{2qs-ix}{2sqrt{s}}right) + text{erfi}left(frac{2qs+ix}{2sqrt{s}}right) right] , ,$$
but in a general case (or even just for $N=2$) I cannot solve the integral. Any ideas?
integration definite-integrals
asked Nov 15 at 19:40
Jens
316
In my research I am interested in finding the following expression:
$$ f{}_omega^N(x) := intlimits_0^{ell^{,2N}} text{d}s ~ mathcal{I}{}_omega^N(x,s) $$
where the main problem lies in finding an analytic expression for
$$ mathcal{I}{}_omega^N(x,s) := intlimits_0^infty frac{text{d}q}{pi} cos(qx) (omega^2-q^2)^{N-1} e^{-s(omega^2-q^2)^N} , , quad N ge 2 , , quad s > 0 , , quad x in mathbb{R} , .$$
For $s>0$ this integral is convergent, and it can be evaluated numerically in principle. However, it would be nice to have an analytic expression.
In the case $N=1$ the result can be specified in terms of the imaginary error function and one obtains
$$ mathcal{I}^1_omega(x,s) = frac{e^{x^2/(4s)-s,omega^2}}{4sqrt{pi s}}left[ text{erfi}left(frac{2qs-ix}{2sqrt{s}}right) + text{erfi}left(frac{2qs+ix}{2sqrt{s}}right) right] , ,$$
but in a general case (or even just for $N=2$) I cannot solve the integral. Any ideas?
integration definite-integrals
integration definite-integrals
asked Nov 15 at 19:40
Jens
316
asked Nov 15 at 19:40
Jens
316
edited Nov 23 at 8:10
asked Nov 15 at 19:40
Jens
316
asked Nov 15 at 19:40
Jens
316
asked Nov 15 at 19:40
Jens
316
316
add a comment |
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