Using power series expansions, find a function $f$ which is holomorphic on the unit disk $D:=$ {$z\in\mathbb C:|z|<1$} and solves the differential equation $(1-z^2)f”(z)-4zf'(z)-2f(z)=0$ for $z\in D$ along with the initial conditions $f(0)=0$, and $f'(0)=1$ I have been given the hint “for any $c_0,c_1 \in\mathbb C$, we have $$\sum_{j=0}^{\infty} (c_0z^{2j} +c_1z^{2j+1}) =\frac{c_0+c_1z}{1-z^2} $$ as long […]

Suppose I have a matrix of the form $$U\ =\ (I+z\thinspace X)^{\frac{1}{2}}$$ where $I$ is the $n\times n$ identity matrix, $z\in\mathbb{C}$ and $X$ is a $n\times n$ arbitrary complex matrix with entries following $|X_{ij}|\le1$. If $z$ has a small modulus ($|z|\ll1$), am I allowed to expand in a power series the square root matrix expression […]

This question already has an answer here: Evaluating $\lim\limits_{n\to\infty} e^{-n} \sum\limits_{k=0}^{n} \frac{n^k}{k!}$ 8 answers

Problem Consider a uniformly bounded sequence over the real line: $$f_n:\mathbb{R}\to\mathbb{C}:\quad|f_n(x)|\leq L$$ Suppose they have analytic continuations with common domain: $$F_n:\Omega\to\mathbb{C}:\quad F_n\restriction_\mathbb{R}=f_n$$ Does their uniform limit have an analytic continuation, too? $$F:\Omega\to\mathbb{C}:\quad F\restriction_\mathbb{R}=f\quad(f_n\stackrel{\infty}{\to}f)$$ (By uniform boundedness this seems very likely; but really?) Application An almost modular state is modular: $$A\in\mathcal{A}^\omega:\quad\omega(\sigma^t[A]B)=\omega(B\sigma^{t+i\beta}[A])\quad(B\in\mathcal{A})$$ (Supposed that entire elements are […]

I can expand $(x+2)^{1/2}$ by taking $y=x+1$ and using the binomial series to find a power series representation of $f$: $$\sqrt{y+1}=\sum_{k=0}^{\infty}{1/2\choose k}y^k=1+\frac{y}{2}+\frac{\frac{1}{2}\frac{-1}{2}y^2}{2!}+O(y^3)=1+\frac{x+1}{2}-\frac{1}{8}(x+1)^2+O(x^3) $$ And grouping terms $$\sqrt{x+2}=(1+1/2-1/8)+\frac{x}{4}-\frac{x^2}{8}+O(x^3)=\frac{11}{8}+\frac{x}{4}-\frac{x^2}{8}+O(x^3)$$ I am not sure how this is supposed to bring me closer to the taylor series representation, and in particular the expansion about $x=2$. Sorry if some of […]

T. Gunn’s answer to a question about linear combinations being restricted to finite sums asserted that For example, you would not count the “infinite” linear combination $$\exp(x) = \sum_{n = 0}^\infty \frac{x^n}{n!} $$ to be in the span of $\{1,x,x^2,\dots\}$ in the vector space $C[0,1]$ of continuous functions on $[0, 1]$. After some brief reading, […]

For a formal power series $$F(x) = \sum p_i x^i$$ a multiplicative inverse of $F$ exists iff $p_0 \neq 0$. The inverse $\sum q_i x^i$ satisfies the recursion $$q_0 =\frac{1}{p_0}\\ q_{n} = -\frac{1}{p_0}\sum_{0 \leq i < n}p_{n-i}q_{i}$$ What’s the closed form of this recurrence? Writing out a bunch of terms hasn’t yet revealed to me […]

I came across the following sum: $\sum_{m=1}^{\log_2(N)} 2^{m}$. My intuition tells me that this should be bounded by 2N, but how would I prove this?

Are there other known power series for the Lambert W function, other than this one: $$W(x) = x-x^2+\frac{3 x^3}{2}-\frac{8 x^4}{3}+\frac{125 x^5}{24}-\frac{54 x^6}{5}+\frac{16807 x^7}{720}-\frac{16384 x^8}{315}+\frac{531441 x^9}{4480}-\frac{156250 x^{10}}{567}+\frac{2357947691 x^{11}}{3628800}-\frac{2985984 x^{12}}{1925}+O\left(x^{13}\right)$$ Series[LambertW[x], {x, 0, 12}] Edit 30.11.2013: Is this a valid generalization to any z? $$W(z) = \text{Log}[z]+\frac{(-2-\text{Log}[z]) (z-z \text{Log}[z])^2}{2 (1+\text{Log}[z]) (z+z \text{Log}[z])^2}+\frac{z-z \text{Log}[z]}{z+z \text{Log}[z]}+\frac{(z-z \text{Log}[z])^3 \left(9+8 \text{Log}[z]+2 […]

I’m not sure how to handle this problem. I got that the radius of convergence was 1/6, but I don’t know how to represent the function as a power series. I can modify it to look like the following: x * 1/(1-(-6x))^2 I thought this would be the power series but apparently not.

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