Intereting Posts

Basis for $\mathbb{R}$ over $\mathbb{Q}$
Is this $\gcd(0, 0) = 0$ a wrong belief in mathematics or it is true by convention?
Proving there don't exist $F(x), G(x)$ such that $1^{-1}+2^{-1}+3^{-1}+\cdots+n^{-1}={F(n)}/{G(n)}$
Proving that $\sum_{k=1}^{\infty} \frac{3408 k^2+1974 k-720}{128 k^6+480 k^5+680 k^4+450 k^3+137 k^2+15 k} = \pi$
Proving $\left(\sum_{n=-\infty}^\infty q^{n^2} \right)^2 = \sum_{n=-\infty}^\infty \frac{1}{\cos(n \pi \tau)}$
What is vector division?
What's the deal with empty models in first-order logic?
Why isn't the volume formula for a cone $\pi r^2h$?
Analog of $(a+b)^2 \leq 2(a^2 + b^2)$
Is there such a thing as a countable set with an uncountable subset?
Distance/Similarity between two matrices
Numbers represented by a cubic form
Can I apply the Girsanov theorem to an Ornstein-Uhlenbeck process?
Does this sequence have any mathematical significance?
Why are fields with characteristic 2 so pathological?

I would like to solve the integral $\int_{\gamma} \frac{1}{z \bar{z}} dz$. Here $\gamma \subset \mathbb{C}$ is a square centered at the origin and where his vertices are parallel to the axes. Could I use the Cauchy theorem in using the fact we will change the contour to a unit circle center to the origin?

- Show that $\int_0^\infty\frac{1}{1+x^n}\,\mathrm dx = \frac{\pi/n}{\sin(\pi/n)}$ for $\mathbb{N}\ni n\geq 2$
- Are there any simple ways to see that $e^z-z=0$ has infinitely many solutions?
- compute integral $\int_0^{2\pi} \frac{1}{z-\cos(\phi)} d\phi$
- Mean-value property for holomorphic functions
- Where does Klein's j-invariant take the values 0 and 1, and with what multiplicities?
- Prove that $\prod_{n=2}^∞ \left( 1 - \frac{1}{n^4} \right) = \frac{e^π - e^{-π}}{8π}$
- Holomorphic Parameter Integral
- erf(a+ib) error function separate into real and imaginary part
- analytic functions defined on $A\cup D$
- Calculate using residues $\int_0^\infty\int_0^\infty{\cos\frac{\pi}2\Big(nx^2-\frac{y^2}n\Big)\cos\pi xy\over\cosh\pi x\cosh\pi y}dxdy,n\in\mathbb{N}$

The integrand is not holomorphic, you can’t use Cauchy integral theorem.

However, if you parametric $\gamma$ by $[0,2\pi] \ni \theta \mapsto z(\theta) = r(\theta) e^{i\theta}$ by a suitable chosen real valued function $r(\theta)$.

The integral becomes

$$\int_{\gamma} \frac{dz}{z\bar{z}} = \int_0^{2\pi} \left( \frac{r'(\theta)+ir(\theta)}{r(\theta)^2} \right) e^{i\theta} d\theta$$

Notice $r(\theta+\pi) = r(\theta)$ while $e^{i(\theta+\pi)} = -e^{i\theta}$. the contribution from $[\pi,2\pi]$ in last integral cancel the one from $[0,\pi]$. So the integral is $0$.

- Find smallest number which is divisible to $N$ and its digits sums to $N$
- For which $x$ is $e^x$ rational? Transcendental?
- Where does the Pythagorean theorem “fit” within modern mathematics?
- If $\mu$ is a probability measure on $\mathbb R$, is $t\mapsto\mu\{t\}$ differentiable almost everywhere?
- Asymptotic of a sum involving binomial coefficients
- Neighborhoods vs Open Neighborhoods?
- Continuity of a monotonically increasing function
- Whats the difference between Antisymmetric and reflexive? (Set Theory/Discrete math)
- Complete first order theory with finite model is categorical
- A problem on limit
- Lemma/Proposition/Theorem, which one should we pick?
- Proving Congruence for Numbers
- A proof that BS(1,2) is not polycyclic
- Doubts about a nested exponents modulo n (homework)
- Bounded inverse operator