# Tagged Questions

33 views

### Solving an integral (using Cauchy contour integral?)

I need to solve this integral: $$f(t)=\int_0^\infty x^2 \sqrt x \left( e^{a x} -1\right)^{-1/2} \frac{e^{i(b-x)t}-1}{b-x} dx$$ where $a$ and $b$ are real, positive ...
45 views

### Applying Green's Theorem to a Closed Complex Contour Integral

How would one apply Green's Theorem to the following complex contour integral: $\oint_\gamma$ $\frac{u^{s-1}}{e^{-u}-1)}du$. Where $\gamma$ is the Hankel Contour (counterclockwise) and R is the ...
62 views

### Contour integral in complex plane (tricky)

Let U be a simply connected domain with a simple closed boundary curve C oriented anticlockwise, and define for all w ∈ C \ C $$g(w)=\oint_C \frac{e^zdz}{(z-w)^2}$$ Find a formula for g(w) which does ...
44 views

### Stokes' Theorem and Surfaces

Stokes' Theorem states the following: \begin{equation*} \oint_c \textbf{F}\centerdot d\textbf{r}= \int\int_S (\nabla \times\textbf{F})\centerdot nd \textbf{S}\end{equation*} for a given C that is the ...
85 views

### Guidance or advice with $I=\int_0^{2\pi}\frac{1}{4+\cos t}dt$

Let \begin{align} I=\int_0^{2\pi}\frac{1}{4+\cos t}dt \end{align} I would like to evaluate this integral using cauchhy's Integral formula, I understand that I have to convert this into a form ...
34 views

### Prove that $\frac{1}{2\pi i}\int_{C_r}\frac{e^{\lambda t}}{\lambda^{k+1}}d\lambda =\frac{t^k}{k!}$

Let $C_r$ be the circle centered on $0$ with radius $r$ and $t\in \mathbb{R}$. How to show that $$\frac{1}{2\pi i}\int_{C_r}\frac{e^{\lambda t}}{\lambda^{k+1}}d\lambda =\frac{t^k}{k!}$$
94 views

### Integrating $e^{a/x^2-x^2}/(1-e^{b/x^2})$

I want to solve the following two integrals analytically \begin{aligned} I_1 = & \int\limits_0^{\infty}\frac{e^{a/x^2}}{1-e^{b/x^2}}e^{-x^2}dx \\ I_2 = & ...
64 views

### What does this complex contour integral represent?

How would one evaluate the following complex contour integral in "Integral and Series Representations of Riemann’s Zeta function, Dirichelet’s Eta Function and a Medley of Related Results." The ...
65 views

### Integrating $xe^{a/x^2 - x^2}\text{Erfi}(x/\sqrt{2})$?

I want to solve any of the two integrals for the complex number $a$ \begin{aligned} I_1 & = \int\limits_{0}^{\infty} xe^{a/x^2 - x^2}\text{Erfi}(x/\sqrt{2}) dx\\ I_2 & = ...
102 views

### Evaluation of tricky integral

I want to evaluate the integral $$\int _ {b} ^ {\infty} \mathrm{d} x \, \frac{e ^ {x ^ {2} / s} (b^2 + 3 x ^ 2) ^ {2}}{x (x^2 + b^2)}$$, where $b$ and $s$ are positive real numbers. I thought of ...
207 views

83 views

### Integral $I=\int_0^1 \frac{\arctan\big(\sqrt{x^2 + 2}\big)}{\sqrt{x^2 + 2}(x^2 + 1)}dx$ [duplicate]

Hi I'm trying to show that $$I=\int_0^1 \frac{\arctan\big(\sqrt{x^2 + 2}\big)}{\sqrt{x^2 + 2}(x^2 + 1)}dx=\frac{5\pi^2}{96}.$$ We can try the substitution $u=(x^2+2)^{1/2}, du=x(2+x^2)^{-1/2}dx$ ...
95 views

### LogSine Integral $\int_0^{\pi/3}\ln^n\big(2\sin\frac{\theta}{2}\big)\mathrm d\theta$

I am trying to integrate the Log Sine Integral: $$Ls_{n+1}=-\int_0^{\pi/3}\bigg[\ln\big(2\sin\frac{\theta}{2}\big)\bigg]^n\mathrm d\theta$$ where $n$ is a non-negative integer. This problem is ...
75 views

### LogSine Generating Fn $\int_0^\pi \big(2\sin\frac{\theta}{2}\big)^x e^{\theta y} d\theta$

This is related to generating functions for Ls (Log Sine Integrals.) I am trying to calculate $$\int_{0}^{\pi}\left[2\sin\left(\theta \over 2\right)\right]^{x} {\rm e}^{\theta y}\,{\rm d}\theta.$$ ...
177 views

166 views

126 views

### Integrate $\int_0^\infty \frac{\sqrt{x}}{e^{(x-\alpha)\beta}+1}dx$

I need to solve for the parameter $\alpha$ after I calculate the integral.$$\mathcal{R}(\alpha,\beta)=\int_0^\infty \frac{\sqrt{x}}{e^{(x-\alpha)\beta}+1}dx, \ \ \beta >0$$ The result of this ...
306 views

### Integrate $\int_0^\infty \frac{ \ln^2(1+x)}{x^{3/2}} dx=8\pi \ln 2$

I am trying to evaluate this integral. $$I=\int_0^\infty \frac{ \ln^2(1+x)}{x^{3/2}} dx=8\pi \ln 2$$ Note $$\ln(1+x)=\sum_{n=1}^\infty \frac{(-1)^{n+1}x^n}{n}, \ |x| < 1.$$ I was trying to do ...
126 views

### erf(a+ib) error function separate into real and imaginary part

Is there an easy way to separate erf(a+ib) into real and imaginary part?
162 views

### Integral $\int_0^a \ln \left( \frac{b-\sqrt{a^2-x^2}}{b+\sqrt{a^2-x^2}} \right)dx$

Hi I am trying to calculate, $$\int_0^a \ln \left( \frac{b-\sqrt{a^2-x^2}}{b+\sqrt{a^2-x^2}} \right)dx$$ where $a,b$ are positive real constants. I Know $\ln(xy)=\ln x +\ln y$, but I do not ...
85 views

### Contour Integral for Cosine and a rational function

I've been trying to figure out this integral via use of residues: $$\int_{-\infty}^{\infty} \displaystyle \frac{\cos{5x}}{x^4+1}dx$$ The usual semicircle contour wont work for this guy as the ...
Find the contour integral of $$\int_C\frac{(z+a)(z+b)}{(z-a)(z-b)} \mbox{d}z,$$ where the modulus of $a$ and $b$ are less than $1$, and the integral path $C$ is the anticlockwise unit circle ($|z|= ... 3answers 424 views ### Integrating$\int_0^\infty \frac{\log x}{(1+x)^3}\,dx$using residues I am trying to use residues to compute $$\int_0^\infty\frac{\log x}{(1+x)^3}\,dx.$$My first attempt involved trying to take a circular contour with the branch cut being the positive real axis, but ... 1answer 726 views ### Integral of$e^{-x^2}\cos(x^2)\$ using residues
I want to solve the following integral: $$\int_0^{\infty} \!\! \operatorname{e}^{-x^2}\!\cos(x^2) \, \operatorname{d}\!x$$ I have seen this in a section about residues, so my guess is that I would ...