Intereting Posts

Explicit example of countable transitive model of $\sf ZF$
Proving limit with $\log(n!)$
A reference for existence/uniqueness theorem for an ODE with Carathéodory condition
Calculate the infinite sum $\sum_{k=1}^\infty \frac{1}{k(k+1)(k+2)\cdots (k+p)} $
Could a square be a perfect number?
$F$ is a field iff $F$ is a Principal Ideal Domain
If $f(0) = f(1)=0$ and $|f'' | \leq 1$ on $$, then $|f'(1/2)|\le 1/4$
reference for finit sum of cotangents
The probability that two vectors are linearly independent.
Differential of a multi-variable map
Integral with quadratic square root inside trigonometric functions
Help understanding the difference between the LLNs and CLT?
A probability question that I failed to answer in a job interview
Two questions for coordinate geometry
Is there a positive function $f$ on real line such that $f(x)f(y)\le|x-y|, \forall x\in \mathbb Q , \forall y \in \mathbb R \setminus \mathbb Q$?

I need to prove that if $f$ and $g$ are analytic in $D_r(z_0)$ and $g$ has a simple zero at $z_0$, then $$\mathrm{Res}[f/g,z_0]=\frac{f(z_0)}{g'(z_0)}$$

When $f(z_0)\neq 0$ and since $1/g$ has a simple pole at $z_0$ $$\mathrm{Res}[f/g,z_0]=\lim_{z\to z_0}(z-z_0)\frac{f(z)}{g(z)}=\frac{f(z_0)}{g'(z_0)}$$

But what do I need to do when $f(z_0)=0$? Any help will be appreciated thanks

- Is this function nowhere analytic?
- Proving the identity $\sum_{n=-\infty}^\infty e^{-\pi n^2x}=x^{-1/2}\sum_{n=-\infty}^\infty e^{-\pi n^2/x}.$
- Calculate $\int_0^\infty {\frac{x}{{\left( {x + 1} \right)\sqrt {4{x^4} + 8{x^3} + 12{x^2} + 8x + 1} }}dx}$
- How would I go about finding a closed form solution for $g(x,n) = f(f(f(…(x))))$, $n$ times?
- Solve $\cos(z)=\frac{3}{4}+\frac{i}{4}$
- Analytic $F(z)$ has $f(z)$ as derivative $\implies$ $\int_\gamma f(z)\ dz = 0$ for $\gamma$ a closed curve

- Topology and analytic functions
- Classify the singularity - $\frac{1}{e^z-1}$
- What is the radius of convergence of $\sum z^{n!}$?
- Order of growth of the entire function $\sin(\sqrt{z})/\sqrt{z}$
- Analysing a set in the complex plane
- Show that $\int_0^ \infty \frac{1}{1+x^n} dx= \frac{ \pi /n}{\sin(\pi /n)}$ , where $n$ is a positive integer.
- Number of fixed points of automorphism on Riemann Surface
- Integration of $\int_0^\infty\frac{1-\cos x}{x^2(x^2+1)}\,dx$ by means of complex analysis
- What is $iav-\log(v)$? Any series expansion or inequality for it?
- Closed form for an infinite series of Bessel functions

When $f(z_0)=0$ the formula is trivially true because the singularity of $f/g$ is removable and the residue is zero.

- For $f \in L^1(\mathbb{R})$ and $y > 0$, we have ${1\over{\sqrt{y}}} \int_\mathbb{R} f(x – t)e^{{-\pi t^2}\over{y}}dt \in L^1(\mathbb{R})$?
- Prove that $f(x)\in \mathbb{Z}$ such that $f(0)$ and $f(1)$ are odd has no integer roots
- Drawing large rectangle under curve
- An application of Yoneda Lemma
- Compact operator on $l^2$
- Finding the limit of $\frac{\sqrt{x}}{\sqrt{x}+\sin\sqrt{x}}$
- Find all polynomials $P$ such that $P(x^2+1)=P(x)^2+1$
- n-tuple Notation
- The simple roots of a polynomial are smooth functions with respect to the coefficients of the polynomial?
- How to prove the sum of squares is minimum?
- Probability of winning the game 1-2-3-4-5-6-7-8-9-10-J-Q-K
- Is the compactness theorem (from mathematical logic) equivalent to the Axiom of Choice?
- Is $2^{1093}-2$ a multiple of $1093^2$?
- Why is empty product defined to be $1$?
- If two limits don't exist, are we sure the limit of sum doesn't exist?