How to apply the method of steepest descent to solve this integral?
I want to reproduce Eq. (11) of this paper. It is the result of solving the integral
$$
int_{0}^{Lambda} text{d}q frac{q (e^{i q r} - e^{-iqr})}{q^2 - x}
$$
(where $Lambda = pi$ and $x = omega/sqrt{Delta k}$, with $omega$ real and $Delta k$ complex). The authors state that they use the Residue theorem and the method of steepest descent, and they assume $r gg 1$. I see that I can write this as a closed curve around the pole $sqrt{x}$, but I have no idea how to apply the method of steepest descent to this integral.
The integral should give, according to the authors,
$$
e^{iomega rRe(sqrt{Delta k})/|Delta k|}e^{-romega |Im(sqrt{Delta k}|)/|Delta k|}.
$$
residue-calculus
asked Dec 12 '18 at 14:50
Kappie001Kappie001
1486
add a comment |
I want to reproduce Eq. (11) of this paper. It is the result of solving the integral
$$
int_{0}^{Lambda} text{d}q frac{q (e^{i q r} - e^{-iqr})}{q^2 - x}
$$
(where $Lambda = pi$ and $x = omega/sqrt{Delta k}$, with $omega$ real and $Delta k$ complex). The authors state that they use the Residue theorem and the method of steepest descent, and they assume $r gg 1$. I see that I can write this as a closed curve around the pole $sqrt{x}$, but I have no idea how to apply the method of steepest descent to this integral.
The integral should give, according to the authors,
$$
e^{iomega rRe(sqrt{Delta k})/|Delta k|}e^{-romega |Im(sqrt{Delta k}|)/|Delta k|}.
$$
residue-calculus
asked Dec 12 '18 at 14:50
Kappie001Kappie001
1486
add a comment |
I want to reproduce Eq. (11) of this paper. It is the result of solving the integral
$$
int_{0}^{Lambda} text{d}q frac{q (e^{i q r} - e^{-iqr})}{q^2 - x}
$$
(where $Lambda = pi$ and $x = omega/sqrt{Delta k}$, with $omega$ real and $Delta k$ complex). The authors state that they use the Residue theorem and the method of steepest descent, and they assume $r gg 1$. I see that I can write this as a closed curve around the pole $sqrt{x}$, but I have no idea how to apply the method of steepest descent to this integral.
The integral should give, according to the authors,
$$
e^{iomega rRe(sqrt{Delta k})/|Delta k|}e^{-romega |Im(sqrt{Delta k}|)/|Delta k|}.
$$
residue-calculus
asked Dec 12 '18 at 14:50
Kappie001Kappie001
1486
I want to reproduce Eq. (11) of this paper. It is the result of solving the integral
$$
int_{0}^{Lambda} text{d}q frac{q (e^{i q r} - e^{-iqr})}{q^2 - x}
$$
(where $Lambda = pi$ and $x = omega/sqrt{Delta k}$, with $omega$ real and $Delta k$ complex). The authors state that they use the Residue theorem and the method of steepest descent, and they assume $r gg 1$. I see that I can write this as a closed curve around the pole $sqrt{x}$, but I have no idea how to apply the method of steepest descent to this integral.
The integral should give, according to the authors,
$$
e^{iomega rRe(sqrt{Delta k})/|Delta k|}e^{-romega |Im(sqrt{Delta k}|)/|Delta k|}.
$$
residue-calculus
residue-calculus
asked Dec 12 '18 at 14:50
Kappie001Kappie001
1486
asked Dec 12 '18 at 14:50
Kappie001Kappie001
1486
asked Dec 12 '18 at 14:50
Kappie001Kappie001
1486
asked Dec 12 '18 at 14:50
Kappie001Kappie001
1486
asked Dec 12 '18 at 14:50
Kappie001Kappie001
1486
1486
add a comment |
add a comment |
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