Integral which might require contour integration
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For a physics problem I need to calculate the following integral:
$$I = int frac{(alpha^2/k^2)}{frac{vec{p}^2}{2}-frac{(vec{p} - vec{k})^2}{2} -1 +ivarepsilon} frac{d^3 vec{k}}{(2pi)^3}
$$
First I rewrite as
$$I = int_{0}^{2pi}int_{0}^{pi}int_{-infty}^{infty} frac{alpha^2 sin(theta) }{frac{k^2}{2} - pkcostheta -1 + ivarepsilon}dkdtheta dphi$$
Where $k = |vec{k}|$ and $varepsilon$ is a positive infinitesimal. I suspect that to calculate this integral I need a contour integral but I'm lost on how to calculate it. $alpha$ is positive and I may assume $alpha <<1$ But I would like to calculate exactly if possible.
integration contour-integration
edited Nov 20 at 21:03
Key Flex
7,44941232
asked Nov 20 at 21:02
Pam
312
add a comment |
up vote
1
down vote
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For a physics problem I need to calculate the following integral:
$$I = int frac{(alpha^2/k^2)}{frac{vec{p}^2}{2}-frac{(vec{p} - vec{k})^2}{2} -1 +ivarepsilon} frac{d^3 vec{k}}{(2pi)^3}
$$
First I rewrite as
$$I = int_{0}^{2pi}int_{0}^{pi}int_{-infty}^{infty} frac{alpha^2 sin(theta) }{frac{k^2}{2} - pkcostheta -1 + ivarepsilon}dkdtheta dphi$$
Where $k = |vec{k}|$ and $varepsilon$ is a positive infinitesimal. I suspect that to calculate this integral I need a contour integral but I'm lost on how to calculate it. $alpha$ is positive and I may assume $alpha <<1$ But I would like to calculate exactly if possible.
integration contour-integration
edited Nov 20 at 21:03
Key Flex
7,44941232
asked Nov 20 at 21:02
Pam
312
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
For a physics problem I need to calculate the following integral:
$$I = int frac{(alpha^2/k^2)}{frac{vec{p}^2}{2}-frac{(vec{p} - vec{k})^2}{2} -1 +ivarepsilon} frac{d^3 vec{k}}{(2pi)^3}
$$
First I rewrite as
$$I = int_{0}^{2pi}int_{0}^{pi}int_{-infty}^{infty} frac{alpha^2 sin(theta) }{frac{k^2}{2} - pkcostheta -1 + ivarepsilon}dkdtheta dphi$$
Where $k = |vec{k}|$ and $varepsilon$ is a positive infinitesimal. I suspect that to calculate this integral I need a contour integral but I'm lost on how to calculate it. $alpha$ is positive and I may assume $alpha <<1$ But I would like to calculate exactly if possible.
integration contour-integration
edited Nov 20 at 21:03
Key Flex
7,44941232
asked Nov 20 at 21:02
Pam
312
For a physics problem I need to calculate the following integral:
$$I = int frac{(alpha^2/k^2)}{frac{vec{p}^2}{2}-frac{(vec{p} - vec{k})^2}{2} -1 +ivarepsilon} frac{d^3 vec{k}}{(2pi)^3}
$$
First I rewrite as
$$I = int_{0}^{2pi}int_{0}^{pi}int_{-infty}^{infty} frac{alpha^2 sin(theta) }{frac{k^2}{2} - pkcostheta -1 + ivarepsilon}dkdtheta dphi$$
Where $k = |vec{k}|$ and $varepsilon$ is a positive infinitesimal. I suspect that to calculate this integral I need a contour integral but I'm lost on how to calculate it. $alpha$ is positive and I may assume $alpha <<1$ But I would like to calculate exactly if possible.
integration contour-integration
integration contour-integration
edited Nov 20 at 21:03
Key Flex
7,44941232
asked Nov 20 at 21:02
Pam
312
edited Nov 20 at 21:03
Key Flex
7,44941232
asked Nov 20 at 21:02
Pam
312
edited Nov 20 at 21:03
Key Flex
7,44941232
edited Nov 20 at 21:03
Key Flex
7,44941232
edited Nov 20 at 21:03
Key Flex
7,44941232
7,44941232
asked Nov 20 at 21:02
Pam
312
asked Nov 20 at 21:02
Pam
312
asked Nov 20 at 21:02
Pam
312
312
add a comment |
add a comment |
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