Articles of complex analysis

Integrability of $f(t) =\frac{2^{\frac{it+1}{1.5}}}{2^{\frac{it+1}{2}}} \frac{\Gamma \left( \frac{it+1}{1.5} \right) }{\Gamma \frac{ it+1}{2} }$

Can we show that the following function is integrable \begin{align} f(t) =\frac{2^{\frac{it+1}{1.5}}}{2^{\frac{it+1}{2}}} \frac{\Gamma \left( \frac{it+1}{1.5} \right) }{\Gamma \left(\frac{ it+1}{2} \right)}, \end{align} where $t \in \mathbb{R}$ and $i=\sqrt{-1}$. That is can we show that \begin{align} \int_{-\infty}^{\infty} |f(t)| dt<\infty. \end{align} I was wondering if Stirling’s approximation can be used, since this is a complex case? Note if […]

Using term-by-term Integration to solve Laplace Transforms

I am attempting to use term by term integration to find the LaPlace transform of $$u(t) = \frac{\sin(t)}{t}H(t)$$ The Laplace transform is going to be $\int_0^\infty \frac{\sin(t)e^{-st}}{t}$. Every series that I try to come up with for this function is an unrecognizable mess. How do I approach this problem? Or do I misunderstand what term-by-term […]

$f(z)=\int_1^\infty e^{-x}x^z\,dx$ is complex analytic

Note: I’m refereshing my complex analysis skills in order to learn some analytic number theory. Here’s one (basic) claim I’d like to prove and my attempt. My questions are: Is my partial attempt correct? Are there better (or shorter) ways to prove it? I may be going a bit too much into the details instead […]

Integral $\int_{0}^{A}\frac{\exp(-2\pi iwx)}{x-i}dx $

Here $A>0$, $w$-real,$\mathtt{i}$-complex. Mathematica gives the answer: $$\frac{1}{2}e^{2\pi w}(\mathtt{i}\pi+2\Gamma(0,2\pi w)-2\Gamma(0,2(1+\mathtt{i}A)\pi w)+2\ln(-\mathtt{i}+A)+2\ln(w)-2\ln(w+iA)) $$ My questions: 1. How to obtain this results without mathematica? 2. Why does this integral grow exponentially as a function of $w$? Mostly I don’t understand why this integral explodes.The real part is:$$\int_{0}^{A}\frac{x\cos(2\pi wx)+\sin(2\pi wx)}{1+x^{2}}\ {d}x $$ and geometrically I don’t understand why […]

Hypotheses in Morera's theorem

I think of Morera’s theorem as saying that if $f$ maps some open subset of $\mathbb C$ into $\mathbb C$ and the integral of $f$ along every simple close curve (or maybe piecewise smooth simple closed curve?) in that open subset is $0$, then $f$ is holomorphic on that subset. Some versions of Morera’s theorem […]

Understanding angle-preserving definition

My book (Real and complex analysis, by Rudin) gives the following definition: Let $A(z) = \frac z{|z|}$. Then we say $f$ preserves angles at $z_0$ if $$\lim_{r \to 0}e^{-i\theta} A[f(z_0 + re^{i\theta} )- f(z_0)] \ \ \ \ (r > 0)$$ exists and it is independent of $\theta$. It then adds The requirements is that […]

Geometric Interpretation of Complex Algebraic Proof of Sum of Squares Statement

When I see answers regarding proofs such as the one mentioned here, it seems that there is a considerable diversity of ways to attempt to look at this proof. Similarly, although this sum of squares question is very different, I was wondering if there was a geometric interpretation to the following problem: Prove that given […]

Find and classify singular points of $\cot\left(\frac{1}{z}\right)$

I need to find and classify singular points (i.e., decide whether the point is removable, a pole of order $N$, essential, or not an isolated singular point), including infinity, of $\cot\left(\frac{1}{z} \right)$. You can do a really cool thing to write out the Laurent series, and from it, it looks as though $\cot\left(\frac{1}{z}\right)$ has an […]

Find the Laurent series and residue of $\frac{z}{(sinz)^2}$ at $z_0 = 0$.

Find the Laurent series for the given function about the indicated point. Also, give the residue of the function at the point. $$\frac{z}{(sin z)^2}\quad \text{at}\quad z_0 = 0 \quad(four\quad terms\quad of\quad the\quad Laurent\quad series)$$ I am not sure how to approach this question. Can anyone help me with this? Thank you.

Finding the singularity type at $z=0$ of $\frac{1}{\cos(\frac{1}{z})}$

I have the following homework problem: What kind of singular point does the function $\frac{1}{\cos(\frac{1}{z})}$ have at $z=0$ ? What I tried: We note (visually) that $z_{0}$ is the same type of singularity for both $f,f^{2}$ hence the type of singularity of $f(z)=\frac{1}{\cos(\frac{1}{z})}$ have at $z=0$ is the same type of singularity $f^{2}(z)=\frac{1}{\cos^{2}(\frac{1}{z})}$. We recall […]