Tagged Questions

27 views

Computing $\int_{\gamma} {dz \over (z-3)(z)}$

Compute, using the Cauchy Integral Formula, $$\int_{\gamma} {dz \over (z-3)(z)}$$ where $\gamma$ is the circle of radius $2$ centered at the origin, oriented counterclockwise. ...
38 views

verifying log rules via elementary properties of an integral

my attempt at this question so far, $$\int_1^x \frac{1}{t}dt +\int_1^y \frac{1}{t}dt= 2\int_1^x \frac{1}{t}dt+ \int_x^y \frac{1}{t}dt$$ But I am not sure hwo to prove it from elementary ...
35 views

Fundamental Theorem of Calculus and inverse..

If $F(x)$ is defined as $$F(x)= \int_{a}^{x} f(t) dt$$ calculate $(F^{-1})'(y)$ in terms of $f$. I have been working on this for a while now, does the aanswer to this incorporate the Inverse ...
678 views

57 views

Fast way to do this well-known integral (gaussian-distribution)

I want to evaluate $$\frac{1}{\sqrt{2 \pi } \sigma}\int_{-\infty}^{\infty} x^2e^{-\frac{(x-\mu)^2}{2\sigma ^2}}dx.$$ The problem is, I don't want to run into heavy calculations. Therefore, maybe ...
107 views

141 views

How to calculate this integral

I have a function $u(x,y)$ defined on the domain $|x|<\infty, y>0$. I know that $$\frac{\partial u(x,y)}{\partial y} = \frac{y}{\pi}\int_{-\infty}^{\infty} \frac{f(w)}{y^2 + (x-w)^2}dw$$ How ...
33 views

fancy about some properties of kernel functions at infinity

Consider the two common types of kernel functions $\sum\limits_{t=a}^bf(t)K(x,t)$ and $\int_a^bf(t)K(x,t)~dt$ , prove whether the following properties are correct or not: $1.$ If $K(x,t)$ is bounded ...
245 views

What is $\int_0^{\infty}\!e^{-x^2}e^{-ae^{bx^2}}\,dx$?

I've been trying without success to evaluate $${\Large\int_0^{\infty}\!e^{-x^2}e^{\Large \,-ae^{\,bx^2}}\,dx}.$$ It's not in my integral tables. Wolfram online integrator won't do it. It doesn't ...
Integral $\int_0^\pi \cot(x/2)\sin(nx)\,dx$
It seems that $$\int_0^\pi \cot(x/2)\sin(nx)\,dx=\pi$$ for all positive integers $n$. But I have trouble proving it. Anyone?