# Tagged Questions

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### showing solution to kummer differential equation

struggling to solve kummer's differential equation and show that the confluent hyper geometric series is a solution. I have simplified the problem to showing that the sum over j to infinity of ...
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### a differential equation equation related to fourier series

I am really struggling with this one. Any help is welcome! For equation $f''(z) + p(z) f'(z) + q(z) f(z) = 0$, where $p(z)$ and $q(z)$ are fixed polynomials. Given $f(0)=f_0$, $f'(0)=f_1$, prove that ...
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### Power series: If $|z-z_0|< R$ the series converges absolutely

I'm trying to prove absolute convergence of the power seris $$\sum_{n=0}^{\infty} a_n (z-z_0)^n, \qquad |z-z_0| < R$$ where $R^{-1} = \limsup |a_n|^{1/n}$. WLOG, suppose $z_0=0$ (otherwise ...
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### $S(t) = \sum_{n=1}^{+\infty}\frac{e^{int}}{n}$

Let $t \in (0, 2\pi)$, calculate $S(t) = \sum_{n=1}^{+\infty}\frac{e^{int}}{n}$. If we can derive term by term is easy, but how to prove that it is possible ?
Let $\{a\}_{n} \subset \mathbb{C}$ , $a_{n} \rightarrow 0$ , $\sum_{n=0}^{+\infty}|a_{n}-a_{n+1}| < +\infty$; then show that $\sum_{n=0}^{+\infty}a_{n}z^{n}$ converges if $|z|\le 1 \$ , $z \ne ... 3answers 58 views ### Prove the Identity For Fringe Patterns Prove the two Identities for$-1 < r < 1$$\sum_{n=0}^{\infty} r^n\cos n\theta =\frac{1-r\cos\theta}{1-2r\cos\theta+r^2}$$ \sum_{n=0}^{\infty} r^n\sin{n\theta}=\frac{r \sin\theta ... 1answer 33 views ### Derivative of exp with definition of differentiability Prove with the definition of differentiability that \exp(z) is differentiable in \mathbb C and (\exp(z))' = \exp(z) for all z \in \mathbb C. I tried: \begin{align*} \frac{\exp(z+h) - ... 1answer 63 views ### How to prove that a complex power series is differentiable I am always using the following result but I do not know why it is true. So: How to prove the following statement: Suppose the complex power series \sum_{n = 0}^\infty a_n(z-z_0)^n has radius of ... 1answer 33 views ### Sum \sum_{n=2}^\infty\frac{a^{n+1}}{n(n-1)}z^{n} Given the power series \sum_{n=2}^\infty\frac{a^{n+1}}{n(n-1)}z^{n} , where a>0, find the radius of convergence and the sum of the series. The radius is \frac{1}{a} , but what about the sum? ... 1answer 32 views ### Complex power series (or not quite so?) I'm stuck with this problem. Any hints are appreciated. It just says \mbox{"For what values of}\ z\ \mbox{is}\quad \sum_{n = 0}^{\infty}\left(z \over 1+z\right)^{n}\quad \mbox{convergent ?} $$... 0answers 35 views ### Where on the border of convergence circle series converges and where diverges? I have power series of \sum\limits_{k=2}^{\infty} (\ln k)^{\alpha} z^k. Alpha is a parameter. I've found the radius of convergence. R = 1. If alpha \geq 0 then series diverges for z from boundary ... 1answer 52 views ### Bound of power series coefficients of a growth-order-one entire function An entire function f(z) satisfies$$|f(z)| \leq A_\varepsilon e^{2\pi(M+\varepsilon)|z|}$$for every positive \varepsilon. I want to show that$$\limsup_{n \to \infty}\ [f^{(n)}(0)]^{1/n} \leq ... 0answers 55 views ### How can we interpret the coefficients of Laurent series? The coefficients of a Taylor series of a function about a given point are related to the nth derivatives of the function at that point. Can we make a similar statement about what the (negative-index) ... 0answers 57 views ### Is\sum_{n=3}^\infty \dfrac{z^n}{n \ln n}$uniform coverge on$\lvert z\rvert \ <1$?? I tried to solve this problem using Cauchy's convergence criterion. Maybe that problem's answer is not uniform converge. But I didn't solve it. Please reply why this power series is not uniform ... 1answer 34 views ### Question about a Maclaurin series of an elementary function The Maclaurin series expansion for$(1+z)^\alpha$is as follows: $$(1+z)^\alpha = 1 + \sum_{n=1}^\infty \binom{\alpha}{n}z^n$$ with $$|z|<1$$ What I don't understand is why is$|z|<1$? 2answers 179 views ### Show that$\sum_{n=0}^\infty r^n e^{i n \theta} = \frac{1- r\cos(\theta)+i r \sin(\theta)}{1+r^2-2r\cos(\theta)}$[closed] Show that $$\sum_{n=0}^\infty r^n e^{i n \theta} = \frac{1- r\cos(\theta)+i r \sin(\theta)}{1+r^2-2r\cos(\theta)},$$ where$0\leq r <1$. Using this, prove that$\sum_{n=0}^\infty r^n ...
In our analysis class today, our teacher wanted us to prove the following theorem, or according to him, known as Abel's theorem: If $\sum\limits_{n=0}^{\infty} a_n (z-z_0)^n$ with \$a_n \in ...