As we know the closed form of $$ {1 \over k} {n \choose k – 1} \sum_{j = 0}^{n + 1 – k}\left(-1\right)^{\, j} {n + 1 – k \choose j}B_{j} = \bbox[10px,border:2px solid #00A000]{{n! \over k!}\, \left[\, z^{n + 1 – k}\,\,\,\right] {z \exp\left(z\right) \over 1 – \exp\left(-z\right)}} $$ where $B_{n}$ the Bernoulli sequence […]

This is Chapter VII, $\S$3, exercise 4, from Conway’s book: A Course in Functional Analysis: Let $X$ be a Banach space and $G\subset \mathbb{C}$ an open subset. We say that $f: G \to X$ is analytic if the limit $$\lim_{h \to 0} \frac{ f(z+h)-f(z)}{h}$$ exist in $X$ for all $z \in G$. Prove that if […]

I have been using this formula which I determined for myself for quite some time now for use in everything from the sgn() function to the Kronecker delta to the ceiling and NINT() functions but haven’t quite pinned down why it works or what its limitations may be. Can anyone elaborate as to why this […]

As evident $f_n=\frac{1}{n!}\frac{d^n}{dx^n}g(x)(at x=0)$.If I use Cauchy’s integral formula to find the $nth$ derivative,then I’m stuck,because there also the derivative crops up while finding the residue.

This is a problem from Complex Variable (Conway’s book) 2nd ed. (Section 4.4) 9. Show that if $f: \mathbb{C}\to\mathbb{C}$ is a continuous function such that $f$ is analytic off $[-1,1]$ then $f$ is an entire function. I already have a solution by Morera’s theorem that split this problem in 5 cases. I think this solution […]

Let $C$ denote the unit circle centered at the origin in Complex Plane What is the value of $$ \frac{1}{2\pi i}\int_C |1+z+z^2 |dz,$$ where the integral is taken anti-clockwise along $C$? 0 1 2 3 What I have answered is 0 because it seems like $f(z)$ is analytic at 0 hence by Cauchy’s Theorem.

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