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- Jun 20, 2014

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Determine the value of the definite integral

$$\int_0^\infty \frac{dt}{(1+t^2)t^{1/2}}$$

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- Thread starter
- Moderator
- #1

- Jun 20, 2014

- 1,896

-----

Determine the value of the definite integral

$$\int_0^\infty \frac{dt}{(1+t^2)t^{1/2}}$$

-----

Remember to read the POTW submission guidelines to find out how to submit your answers!

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- Jun 20, 2014

- 1,896

The only singularity inside the contour is at the fourth root of $-1$ at $z_0 = \dfrac{1+i}{\sqrt2}$. The residue there is $\dfrac2{4z^3}$ evaluated at $z_0$, namely $\dfrac{\sqrt2}{2(-1+i)}.$ So Cauchy's theorem tells us that $(1-i)J = \dfrac{2\pi i\sqrt2}{2(-1+i)}$. Therefore $$J = \frac{\pi i\sqrt2}{- (1-i)^2} = \frac{\pi i\sqrt2}{2i} = \frac{\pi}{\sqrt2}.$$

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