Articles of complex analysis

Find the Laurent series for $f(z)=\frac{2}{(z-4)}-\frac{3}{(z+1)}$

The question is 2 parts – I’m to find the Laurent series valid for $$1 < |z| <4$$ and $$|z| > 5$$ I’ve already solved the first part, but I have a conceptual question about the second part. My result for the first part: I found that after putting the $f(z)$ into the standard geometric […]

Can Runge's approximating rat. fns. be required to take certain prescribed values?

Suppose $f$ is analytic on an open set $U$ containing the compact set $K$, and $\{r_n\}$ is a sequence of rational functions provided by Runge’s theorem (having poles in some prescribed set $A$). For a given finite set $\{x_1,\ldots,x_k\}\subset K$, can we additionally demand that for each $n$, $r_n(x_1)=f(x_1),r_n(x_2)=f(x_2),\ldots,r_n(x_k)=f(x_k)$? If yes, can we also require […]

To prove $\left|\frac{p_n(z)}{q_m(z)}\right|\leq \frac{M}{|z|^{m-n}}$ for some $M>0$

To prove there exist $M>0$ and $a_0>0$ such that for $|z|>a_0$, $$\left|\frac{p_n(z)}{q_m(z)}\right|\leq \frac{M}{|z|^{m-n}}$$ where $p_n$ and $q_m$ are the polynomials of degree $n$ and $m$ respectively with $n<m$. I encountered this question in this post. According to the hint, it is enough to show that for large enough $R$ and for every $|z|>R$, we have […]

Power series for complex exponential

Let $z=x+iy$ where $x,y\in\mathbb{R}$. The exponential function is $$e^z=e^x(\cos{y}+i\sin{y}).$$ Using the power series of $e^x$, $\cos{y}$ and $\sin{y}$, find a power series representation for $e^{z}$. Recall that $$e^{x} = \sum_{k=0}^\infty \frac{x^k}{k!},\quad \cos{y} = \sum_{k=0}^\infty \frac{(iy)^{2k}}{(2k)!},\quad\sin{y} = \sum_{k=0}^\infty \frac{i^{2k}y^{2k+1}}{(2k+1)!}$$ Then \begin{align} e^{z} &= \left[\sum_{k=0}^\infty \frac{x^k}{k!}\right]\left[\sum_{k=0}^\infty\left(\frac{(iy)^{2k}}{(2k)!}+i\frac{i^{2k}y^{2k+1}}{(2k+1)!}\right)\right]\\ &= \left[\sum_{k=0}^\infty \frac{x^k}{k!}\right]\left[\sum_{k=0}^\infty\left(\frac{(iy)^{2k}(2k+1)}{(2k+1)!}+i\frac{(iy)^{2k}y}{(2k+1)!}\right)\right]\\ &= \left[\sum_{k=0}^\infty \frac{x^k}{k!}\right]\left[\sum_{k=0}^\infty\frac{(iy)^{2k}}{(2k+1)!}\left(2k+1+iy\right)\right] \end{align} Now I can’t really […]

Let C be a circle. Show that the only subset of C homeomorphic to a circle is C itself

I am trying to answer the question stated in the title. The hint in my book says to realize that for any z on the circle C{z} is still connected. I believe I can deal with case that shows that C{z} is not homeomorphic to any circle, but I am not sure how to generalize […]

solve $\sin(z)=-1$ in the set of complex numbers

I’m pretty sure I’m on the right track, but am I missing anything? Can anything further be done with this? Solve $\sin(z)=-1$ in the set of complex numbers. $\sin(z)=-1$ $\Rightarrow{e^{iz}-e^{-iz} \over 2i} =-1$ $\Rightarrow e^{iz}-e^{-iz} =-2i$ $\Rightarrow e^{iz}-{1 \over e^{iz}} + 2i =0$ $\Rightarrow (e^{iz})^2-1 + 2ie^{iz} =0$ for simplicity say $x=e^{iz}$ $\Rightarrow x^2-2xi-1 =0$ […]

Prove that the taylor series of cos(z) and sin(z) are holomorphic

I have a question on an old exam paper given as follows: If we define $$\cos(z) = 1 – \frac{z^2}{2!} + \frac{z^4}{4!} – … \frac{z^{2n}}{(2n)!} + … = \sum_{n=0}^{\infty} \frac{(-1)^nz^{2n}}{(2n)!}$$ $$\sin(z) = z – \frac{z^3}{3!} + \frac{z^5}{5!} – … \frac{z^{2n+1}}{(2n+1)!} + … = \sum_{n=0}^{\infty} \frac{(-1)^nz^{2n+1}}{(2n+1)!}$$ Then: a) Prove that both series converge in the whole […]

Integrating $\frac {e^{iz}}{z}$ over a semicircle around $0$ of radius $\epsilon$

I am trying to find the value of $\int_{-\infty}^{\infty} \frac{\sin (x)}{x}$ using residue theorem and a contour with a kink around $0$. For this, I need to find $\int_{C_\epsilon} \frac {e^{iz}} {z}$ where $C_\epsilon$ is the semicircle centred at $0$ with radius $\epsilon$ from $-\epsilon$ to $\epsilon$. I guess it is equal to half the […]

Complex number condition on the modulus

The set of all complex numbers $(z_1,z_2)$ which satisfy $$\frac{|z_1 -z_2|}{|1-\overline{z_1}z_2|} \lt 1 $$ is? (Here $\overline{z_1}$ is $z_1$’s cojugate.) I attempted to write $z_1$ a as $x_1 + iy_1$ and $z_2$ as $x_2 + iy_2 $ and tried to simplify. But then it didn’t seem to work. How do I go about this?

expansion of $\text{ cosh}(z+1/z)$

It is given that if $|z|>0$, $$\text{cosh}(z+1/z) = c_0 + c_1 (z + 1/z) + c_2(z^2 + 1/z^2) + \dots $$ where $$c_n = \frac{1}{2\pi}\int_0^{2\pi} \cos (n \phi) \text{cosh}(2\cos \phi ) d\phi$$ I only know to expand $\text{cosh(z + 1/z)}$ by putting $w = z + 1/z$ and expand it as $\frac 12 (e^{w} + […]