How can evaluate $\int_{-1}^{1} z^{\frac{1}{2}}\, dz$ with the main branch of $z^{\frac{1}{2}}$? Thanks for your help
how evaluate $\int_{-1}^{1} z^{\frac{1}{2}}\, dz$?
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0It is being evaluated in the complex plane – 2012-05-07
1 Answers
This is an expansion on anon's comment above.
Caveat: I'm not 100% certain what the "main branch" is supposed to do to the negative real axis, but I am going to assume it maps to the positive imaginary axis.
To integrate from $0$ to $1$, that's no problem, that's an old-school integral of a real-valued function on the real line, and we get 2/3.
From $-1$ to $0$, we have a complex valued function. I think the easiest way to do this one is to let $t = -z$. Now, because you're working with the main branch, $\sqrt{-t} = i\sqrt{t}$ for $t$ a positive real number - note, confusingly, that this identity isn't true for all complex numbers, moreover, a different choice of branch cut of the square root function can make it false. $ \int_{-1}^0 z^{\frac{1}{2}}dz = -i\int_1^0 t^{\frac{1}{2}}dt $ This latter integral is $\frac{2}{3}i$ so the final answer is $\frac{2}{3} + \frac{2}{3}i$.
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1Agreed. Unless a specific curve is given, use this simplest one. For a more complex (ha ha) problem, use a curve that starts at $-1$, wraps around $0$ some number of times, and ends at $1$. Of course the "main branch" is discontinuous when your curve crosses the cut. – 2012-05-07