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for the integral $\int_a^b \frac{\sqrt{(x-a)(b-x)}}{x} dx$, I am thinking use the contour

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then since origin is not inside the contour, so we have $\int_{c-} + \int_{c_b} + \int_{c+} + \int_{c_a} \frac{\sqrt{(z-a)(b-z)}}{z} dz = 0$, but I dont know how to write out the integrals, is $I = \int_{c+} \frac{\sqrt{(z-a)(b-z)}}{z} dz$ and $\int_{c-} \frac{\sqrt{(z-a)(b-z)}}{z} dz = e^{i \pi} I$ ? and what about the two circle contour $\int_{c_b} +\int_{c_a} \frac{\sqrt{(z-a)(b-z)}}{z} dz $ thank you for any help

Ziqin He
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    Not your contour question but are you aware of the nice real integration methods presented here: https://math.stackexchange.com/q/3343625 ? – Jean Marie Oct 07 '21 at 09:07
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    A point of vacabulary that can be helpful for a google query : these types of contours bear the name "dumbbell" or "dogbone". – Jean Marie Oct 07 '21 at 09:23

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