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For $b>0,|a|\le 1$, find $\int_{0}^{\infty}{x^a\over (x+b)^2}dx$. Hint: use Pac-man curve.

My problem is: I have no idea what a Pacman Curve it, and there seem to be no full explanation. The only thing that comes to my mind is using some closed contour that simply avoids $z=-b$ by just halfway encircling it from above or from below. My bigger problem is: what complex function should one take and how does one find the right function to satisfy the above? Is it simply $z^a\over (z+b)^2$? Because I was surprised many times to learn that appropriate functions are not easily come up with. I could really use some guidance.

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    Do you know what a keyhole contour is? – 2017-02-01
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    I do not know such by name, but it sounds like what I described... I guess..I will check. – 2017-02-01
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    Pac man: http://i.stack.imgur.com/UIBMS.png – 2017-02-01
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    Well it makes a lot of sense right now, doesn't it... – 2017-02-01
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    Which is another name for a [keyhole contour](https://en.wikipedia.org/wiki/Methods_of_contour_integration#Example_4_.E2.80.93_branch_cuts). – 2017-02-01
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    I will try to work with that, I appreciate both your guidance so far., – 2017-02-01
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    Glad to help ! :-) – 2017-02-01
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    And yes, you use $\dfrac{z^a}{(z+b)^2}$. Note you must have $-1 < a < 1$ for the integral to exist. – 2017-02-01
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    If you need one last hint, the straight part of the lower jaw of Pac man is given by:$$\int_\infty^0 \cfrac{(z e^{2\pi i})^a}{(z e^{2\pi i}+b)^2}\ dz$$and for simplificatipns, use complex definition of sine. – 2017-02-01

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