fredoniahead
- 18
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Hi All:
I am new to the site, so I thought this would be a good time to post an interesting integral I ran across that I am having a time with. It is a miscellaneous problem in Schaum's Outline of Complex Variables, #86 in ch. 7. I have been self-teaching a little CA when I get time and this one is a little more challenging than others I have encountered.
[math]\int_{0}^{\infty}\frac{1}{(4x^{2}+\pi^{2})\cosh(x)}dx=\frac{\ln(2)}{2\pi}[/math]
Does anyone have a good idea of how to approach this one?.
At first, I thought perhaps a rectangular contour may do the trick, but now I am thinking perhaps a semicircular one may be in order.
It would appear there is a double pole at [math]\frac{\pi i}{2}[/math].
Using the Laurent series, I think the residue is [math]\frac{-i}{4\pi^{2}}[/math]
Which gives [math]2\pi i\left(\frac{-i}{4\pi^{2}}\right)=\frac{1}{2\pi}[/math]
But, how to get that ln(2)?. Maybe use [math]\cos(ix)[/math] in some respect?.
I am new to the site, so I thought this would be a good time to post an interesting integral I ran across that I am having a time with. It is a miscellaneous problem in Schaum's Outline of Complex Variables, #86 in ch. 7. I have been self-teaching a little CA when I get time and this one is a little more challenging than others I have encountered.
[math]\int_{0}^{\infty}\frac{1}{(4x^{2}+\pi^{2})\cosh(x)}dx=\frac{\ln(2)}{2\pi}[/math]
Does anyone have a good idea of how to approach this one?.
At first, I thought perhaps a rectangular contour may do the trick, but now I am thinking perhaps a semicircular one may be in order.
It would appear there is a double pole at [math]\frac{\pi i}{2}[/math].
Using the Laurent series, I think the residue is [math]\frac{-i}{4\pi^{2}}[/math]
Which gives [math]2\pi i\left(\frac{-i}{4\pi^{2}}\right)=\frac{1}{2\pi}[/math]
But, how to get that ln(2)?. Maybe use [math]\cos(ix)[/math] in some respect?.