Questions tagged [branch-points]

A branch point is a point in the complex that can map from a single point to multiple points in the range.

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Branch points of a complicated function

I would like to study the branch points of the function $q(z)=\frac{z^{\frac{1}{3}}+\frac{1}{2}z^{\frac{-1}{3}}}{[1+z^{-\frac{2}{3}}]^{\frac{1}{2}}}$ . I plotted its real part with Mathematica and it ...
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Are contours allowed to contain branch points / branch cuts on the contour (not inside)

I just wanted to verify something, since I saw some notes online that confused me. They were doing some contour integral and $0$ was the only branch point of the function, and they chose the branch ...
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How to Prove That Branch Points Of $\log (f(z))$ are Zeros and Poles of $f(z)$?

In many books and notes, It is written that the branch points of $\log (f(z))$ are points $z\in \Bbb C$ such that $f(z)=0$ or $z$ is a pole of $f$. Here we assuming that zero of $f$ is the sense of ...
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Is infinity a branch point of $\sin(\sqrt{z})$?

Does the function $f(z) = \sin(\sqrt{z})$ have a branch point at infinity? I'm confused because infinity is an essential singularity of $\sin(z)$, so I'm not sure how to do the usual $z\to w=1/z$ ...
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how can I find the branch points and cut?

For $f(z)=\sqrt{z^3+8}$, I need to find all branch points and branch cuts, not in the disk $|z|<2$, which make the function single-valued. Here I started with $z=-2,1-i\sqrt{3},1+i\sqrt{3}$ are ...
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How can we know if a solution of differential equation is multivalued without solving it?

For a differential equation, say $q(x)y''(x)+p(x) y'(x)+s(x)=0$, how could we know if its solutions are multivalued without solving it. Could we study the branch points and branch cut by simply ...
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How to solve this integral with contour integration $\int ^{\infty }_{0}\frac{\ln( x)}{( x+1)^{\alpha}}\,dx$

I know how to solve this integral without contour integration. The answer to the integral is $$\int^{\infty }_{0}\frac{\ln( z)}{( z+1)^{\alpha}} \,dx=\frac{H_{\alpha-2}}{1-\alpha} ,\; \alpha>1,$$ ...
I thought about considering $L(z)=log|\frac{1}{z}|+i\theta$ and consider the behaviour as $z$ tends to $0$? All I am observing now is that the function tends to infinity with it. I am really stuck on ...