4
votes
1answer
384 views

Identity involving Fresnel integrals

In the paper E. Mehlum, Appell and the apple (nonlinear splines in space), Technical Report No. 1676 (1981), Central institute for industrial research, Oslo (reproduced in the book Mathematical ...
4
votes
1answer
415 views

Is there a closed form expression/series expansion for $\int_{\epsilon-i\infty}^{\epsilon+i\infty} e^{az+b^2z^2}\Gamma(z)\Gamma(1-z)dz$ ?

I've been trying to find a closed form expression/series expansion for the following integral without success: $$F(a,b)=\int_{\epsilon-i\infty}^{\epsilon+i\infty} ...
8
votes
2answers
481 views

Rain droplets falling on a table

Suppose you have a circular table of radius $R$. This table has been left outside, and it begins to rain at a constant rate of one droplet per second. The drops, which can be considered points as they ...
0
votes
0answers
182 views

Evaluating the integral $\int_{a}^{+\infty} \frac{\exp(-bx)}{x+c} Ei(x) dx$

I'm trying to evaluate or simplify this integral: $$I_{a,b,c} = \int_{a}^{+\infty} \frac{\exp(-bx)}{x+c} Ei(x) dx $$ with $a,b,c \in \mathbb{R}_+^*$. and $ Ei(x) =\int_{-\infty}^{x} ...
0
votes
1answer
229 views

derivative of a special function in integral form

What is the derivative of $Q_m\left(\frac{\alpha}{x^a},\frac{\beta}{x^b}\right)$ with respect to $x$, i.e, $$\frac{\partial}{\partial x}Q_m\left(\frac{\alpha}{x^a},\frac{\beta}{x^b}\right), \quad ...
2
votes
1answer
330 views

A Bessel integral

Today I came across the integral $\int_a^\infty e^{-bx} I_n(x) dx$ where $I_n$ is the modified Bessel function of the first kind. There is a solution for $a=0$, provided in Gradshteyn and Ryzhik, ...
1
vote
1answer
155 views

Integrating with sub-level sets

This is a simple question, and I'm sure it was a homework assignment at some point (assuming it's true) but it's one that I'm puzzled over. Suppose I have a compact domain $D \subset \mathbb{R}^n$ ...
1
vote
3answers
1k views

How do you calculate the solid angle of a rectangular, axis aligned section of a surface defined by a two dimensional function?

I have $f(x,y) = \frac{1}{2} (1 - x^2 - y^2)$, which is a paraboloid centered around the origin (plot). Now I want to calculate the solid angle (with the origin as the viewpoint) of the surface area ...
2
votes
1answer
449 views

Integral and limit

During my research this integral has shown up $ \frac{1}{2T} \int_{-T}^T \left( 1 - \frac{|\tau|}{T}\right)e^{-\alpha\tau^2}\cos(2\pi f_0 \tau) d\tau$ I tried to solved by taking the real part of a ...
0
votes
1answer
904 views

Time scale calculus vs Lebesgue–Stieltjes calculus

About the same time, it seems, as I asked this question, a new post appeared on the wikipedia discussion page for Time scale calculus which suggests the Time scale derivative (aka Hilger derivative ...
0
votes
0answers
479 views

Integrating the product of two functions one of which has a positive non-integer power

I'm looking to integrate several functions having the form $\int_0^T \frac{ sin(\omega \tau) }{\omega} \tau^{2H} d\tau$ where $2H \ge 0$ but may not be an integer. I'd like to know if the machinery ...
4
votes
1answer
2k views

Inverse of a function defined by an integral

Hi, I have a function defined by an integral as follows. $$ z=f(w) = \int_0^w \frac{(\zeta-a_1)^{\alpha_1}(\zeta-a_2)^{\alpha_2}...}{(\zeta-b_1)^{\beta_1}(\zeta-b_2)^{\beta_2}...}\ d\zeta $$ where $w$ ...
0
votes
1answer
940 views

Modified Dirichlet function Darboux integrable on [0,2]?

Hi, Given this modified Dirichlet function: f(x) = 0 if x is in Q, else f(x) = x. I am wondering if this function is Darboux integrable on the interval [0, 2]. I managed to show that every lower ...