2
votes
2answers
65 views

Find $\int_\gamma \frac{dz}{z^2}$ wihtout explicit calculations

Evaluate the following integral without doing any explicit calculations: $\int_\gamma \frac{dz}{z^2}$ where $\gamma(t) = \cos(t) + 2i\sin(t)$ for $0 \le t \le 2\pi$. This exercise comes along with ...
1
vote
2answers
99 views

$\int_{-\infty}^\infty \frac{e^{ax}}{1+e^x}dx$ with residue calculus

I'm trying to compute $\displaystyle \int_{-\infty}^\infty \frac{e^{ax}}{1+e^x}dx$, $(0<a<1)$ Let $f$ denote the integrand. I'm using the rectangular contour given by the following curves: ...
0
votes
1answer
38 views

Integrating Real Function in the Complex Plane

Question: Evaluate the integral $$\int_{-\infty}^{\infty}\frac{\sin(x)}{x(x^2+a^2)}=Im\left ( \frac{e^{ix}}{x(x^2+a^2)} \right)$$ ...
6
votes
3answers
129 views

Show that $\int_0^\infty e^{-x^2} \sin{2xb}\, dx =e^{-b^2}\int_0^b e^{x^2} \, dx, \, b>0, $?

Show that $$\int_0^\infty e^{-x^2} \sin{2xb}\, dx =e^{-b^2}\int_0^b e^{x^2} \, dx, \, b>0, $$ I need help. I did the following steps: Apply Cauchy's Theorem, being $\varphi (x) = e^{-z^2}$ analytic ...
0
votes
1answer
95 views

Complex double integral

I'm having trouble calculating following (complex) integral. $$\int_D z^n dA$$ where $D=\{ z \in \mathbb{C} \mid \lvert z \rvert \leq 1 \}$. I know how to calculate complex (line) integrals and real ...
0
votes
2answers
1k views

How to integrate complex exponential??

Consider $$\int^{\frac{1}{2}}_{-\frac{1}{2} } e^{i2\pi f} \,df = \int^{\frac{1}{2} }_{-\frac{1}{2} } \cos(2 \pi f)\, df$$ Why do we only look at the real part? What about the imaginary part ...
0
votes
1answer
116 views

How to do complex integration. E.g. $ \int_\frac{\pi}{2}^{\frac{\pi}{2} + i} \cos(2z) \; \mathrm{d}z $

For my homework assignment I've been given a number of complex integrals to solve. I've already asked for help on a specific example here, but I was somewhat dissatisfied with the answers. The answers ...
3
votes
1answer
345 views

Evaluating real integral using residue calculus: why different results?

I have to evaluate the real integral $$ I = \int_0^{\infty} \frac{\log^2 x}{x^2+1}. $$ using residue calculus. Its value is $\frac{\pi^3}{8}$, as you can verify (for example) introducing the function ...
3
votes
1answer
475 views

integral of complex logarithm

Consider the integral $$I=\int_0^{2\pi}\log\left|re^{it}-a\right|\,dt$$ where $a$ is a complex number and $0<r<|a|$. We have ...
2
votes
1answer
203 views

line integral versus complex integral

Let $a\in \mathbb C, r>0$ and $\gamma_r=\partial D(0,r)$. I want to evaluate the following line integral $$I=\int_{\gamma_r}\frac{1}{|z-a|^2}ds.$$ I'm looking for a complex function $g(z)$ such ...