Intereting Posts

Integers as a sum of $\frac{1}{n}$
How to solve $n < 2^{n/8}$ for $n$?
Show that the Lie algebra generated by x, y with relations $ad(x)^2(y) = ad(y)^5(x) = 0$ is infinite dimensional and construct a basis
What are the common abbreviation for minimum in equations?
Closure of image by polynomial of irreducible algebraic variety is also irreducible algebraic variety
How to evaluate the following integral using hypergeometric function?
Extension of Sections of Restricted Vector Bundles
operations on probability distributions
Extending a morphism of schemes
topological structure on smooth manifolds
Why is a statement “vacuously true” if the hypothesis is false, or not satisfied?
Induction proof: n lines in a plane
De Morgan's Law – Proof of $(\cup_{i} A_{i})^c = (\cap_{i} A_{i}^c)$
$\mathbb S_n$ as semidirect product
Help with a prime number spiral which turns 90 degrees at each prime

A proof I’m reading tries to evaluate the integral (where $i$ is the regular imaginary unit)

$$\int_{-\infty}^{\infty} e^{-(x-\alpha i)^2}\mathrm{d}x$$

by doing a substitution $u=x-\alpha i$. Normally, one would also have to change the bounds of integration.

- Show that $\int_0^ \infty \frac{1}{1+x^n} dx= \frac{ \pi /n}{\sin(\pi /n)}$ , where $n$ is a positive integer.
- Infinite Series $\sum\limits_{n=1}^\infty\frac{(H_n)^2}{n^3}$
- How to do contour integral on a REAL function?
- Why does a meromorphic function in the (extended) complex plane have finitely many poles?
- What is the reflection across a parabola?
- Recursion relation for Euler numbers

$$\int_{-\infty+\alpha i}^{\infty+\alpha i} e^{-x^2}\mathrm{d}x$$

But this proof leaves the bounds as +/- infinity.

$$\int_{-\infty}^{\infty} e^{-x^2}\mathrm{d}x$$

Why is this valid?

- Evaluate $\sum_0^\infty \frac{1}{n^n}$
- Can I conjugate a complex number : $\sqrt{a+ib}$?
- Is there a geometric insight from this exercise?
- Expressing the area of the image of a holomorphic function by the coefficients of its expansion
- Complex Zeros of $z^2e^z-z$
- Integral with Undefined Endpoint (Complex Variables)
- Approximating roots of the truncated Taylor series of $\exp$ by values of the Lambert W function
- Looking for closed-forms of $\int_0^{\pi/4}\ln^2(\sin x)\,dx$ and $\int_0^{\pi/4}\ln^2(\cos x)\,dx$
- what are the possible values for integral
- On the integral $\int_0^1\frac{dx}{\sqrtx\ \sqrt{1-x}\ \sqrt{1-x\,\gamma^2}}=\frac{1}{N}\,\frac{2\pi}{\sqrt{2\gamma}}$

Consider

$$\oint_C dz \, e^{-z^2}$$

where $C$ is a rectangle having vertices $-R$, $R$, $R+i \alpha$, $-R+i \alpha$. By Cauchy’s Theorem, this integral is zero. On the other hand, it is also equal to

$$\int_{-R+i \alpha}^{R+i \alpha} dx \, e^{-x^2} + i \int_{\alpha}^0 dy \, e^{-(R+i y)^2} -\int_{-R}^R dx \, e^{-x^2} -i \int_0^{\alpha} dy \, e^{-(-R + i y)^2} $$

As $R\to\infty$, the 2nd and 4th integrals vanish because each integral is bounded by the value

$$e^{-R^2} \int_0^{\alpha} dy \, e^{y^2} \le |\alpha| e^{-(R^2-\alpha^2)}$$

Thus, we are left with the equality

$$\int_{-\infty}^{\infty} dx \, e^{-x^2} = \int_{-R+i \alpha}^{R+i \alpha} dx \, e^{-x^2}$$

as was to be shown.

- What are large cardinals for?
- Non-probabilistic proofs of a binomial coefficient identity from a probability question
- Loop space suspension/adjunction
- Order of conjugate of an element given the order of its conjugate
- Fourier transform convention: $\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(x)e^{\pm ikx}dx $?
- Evaluating $\int_0^{\infty}\frac{\ln(x^2+1)}{x^2+1}dx$
- Probability concerning a 6-digit password
- Argument of the Riemann zeta function on Re(s)=1
- Taking the automorphism group of a group is not functorial.
- How many times more than $0$?
- Subset of a finite set is finite
- If $\lim a_n = L$, then $\lim s_n = L$
- How to find $n+1$ equidistant vectors on an $n$-sphere?
- How can I derive this expression related to the triangle inequality?
- Which of the choices solution of the Cauchy problem?