Articles of branch cuts

difficulty understanding branch of the logarithm

Here is one past qual question, Prove that the function $\log(z+ \sqrt{z^2-1})$ can be defined to be analytic on the domain $\mathbb{C} \setminus (-\infty,1]$ (Hint: start by defining an appropriate branch of $ \sqrt{z^2-1}$ on $\mathbb{C}\setminus (-\infty,1]$ ) It just seems typical language problem. I do not see point of being rigorous here. But I […]

Derive branch cuts for $\log(\sqrt{1-z^2} + iz)$ as $(-\infty,-1)$ and $(1,\infty)$?

Attempt: First, we examine $\sqrt{1-z^2}$. Note that it can be written $\sqrt{1-z}\sqrt{1+z}$, so the appropriate branch cuts are $(-\infty,-1)$ and $(1,\infty)$ for the inner square root term. Next, we look at $\log(w)$ and note that we can define the cut for $\log(w)$ as $(-\infty,0)$. But now what? I tried setting $w= \sqrt{1-z^2} + iz$, solving […]

Contour integral of $\sqrt{z^{2}+a^{2}}$

Suppose $a$ is real and nonnegative. Say we wanted to compute the above function (for whatever reason, be it to solve an improper real integral, or something else) along the curve $C$, as on the picture. I have chosen the contour as to avoid the branch cut connecting the three branch points. Supposing $arg\left ( […]

Roots and Logarithms of Matrices.

Ok, this question may be too broad or fuzzy, if it is please let me know and I’ll try and sharpen or narrow it down a bit. Hi. I am aware of some of the difficulties of defining roots and logarithms of matrices, often there are several or even infinite number of ways to do […]

About the branch-cut in the complex logarithm

Say I have the function $log (f(z))$, then does the imaginary part of the value go down by $-i 2 \pi$ on crossing the branch-cut of the log function even if $f(z)$ is not crossing the branch-cut? Specifically, consider the function $log(z^2 +a ^2)$ where $a \in \mathbb{R}$. And I am choosing the branchcuts to […]

Choosing parametrization for complex integration with two branch cuts

I am particularly interested in how Ron Gordon came up with the parametrization in his anser to this question: Inverse Laplace transform $\mathcal{L}^{-1}\left \{ \ln \left ( 1+\frac{w^{2}}{s^{2}}\right ) \right \}$ EDIT: Do we have to incorporate four branch cuts, if for example we wanted to integrate the function $$f\left ( z \right )=\frac{\sqrt{z}}{\ln\left ( […]

Are multi-valued functions a rigorous concept or simply a conversational shorthand?

In Brown and Churchill’s book, the concept of multivalued functions is not discussed in a very rigorous way (if at all). But I can see that branch cuts have importance in complex analysis, so I want to clarify my understanding of multivalued functions. Is there a rigorous development of the definition of a multivalued function […]

An arctan series with a parameter $\sum_{n=1}^\infty \arctan \left(\frac{2a^2}{n^2}\right)$

I’m trying to evaluate $$\sum_{n=1}^\infty \arctan \left(\frac{2a^2}{n^2}\right) \ , \ a >0. $$ The answer I get only seems to be correct for small values of $a$. What accounts for this? Using the principal branch of the logarithm, I get $$ \begin{align} & \sum_{n=1}^\infty \arctan \left(\frac{2a^2}{n^2}\right) \\ & = \text{Im} \sum_{n=1}^\infty \log \left( 1 + […]

Contour Integral $ \int_{0}^1 \frac{\ln{x}}{\sqrt{1-x^2}} \mathrm dx$

I need help evaluating this with contour integration $$ \int_{0}^{1}{\ln\left(\,x\,\right)\over \,\sqrt{\vphantom{\large A}\,1 – x^{2}\,}}\,{\rm d}x $$ I am not sure as to how to work with the branch cuts of both $\ln\left(\,x\,\right)$ and $\sqrt{1 – x^{2}}$ Second part is to evaluate $$ \int_{0}^1 \frac{\sqrt{\,\vphantom{\large a}\ln\left(\,x\,\right)}} {\sqrt{\vphantom{\large A}\,1 – x^{2}\,}} \mathrm dx$$

how to find the branch points and cut

For $f(z) = \sqrt{z^2+1}$, how can I find the branch points and cuts? I took $z=re^{i\theta+2n\pi}$ and substitute in $f(z)$ $$\sqrt{r^2 e^{i(2\theta +4n\pi)}+e^{i 2k\pi}}=$$ then, I don’t know how to deal with this any more and by guessing, I think the branch points should be $i,-i$ and cut is $[-i,i]$