# Tagged Questions

For questions about integration methods that use results from complex analysis and their applications

36 views

### contour integral z/conjugate(z)

I am trying to calculate: $$\int_C \frac{z}{\bar{z}}dz$$ where C is the upper semicircles of the circles centred in (0,0) of radii 1,2 joined at their intersections with the real axis by the real ...
42 views

39 views

25 views

86 views

### Mean Value Property to show that entire function is a constant

Let $f(z)$ be an entire function so that, $$\int \frac{|f(z)|}{1 + |z|^3} dA(z) < \infty$$ where the integral is taken over the entire complex plane. Show that $f$ is a constant. I believe ...
95 views

### Evaluate $\int_0^{\infty} \frac{\log(x)dx}{x^2+a^2}$ using contour integration

This question is Exercise 10 of Chapter 3 of Stein and Shakarchi's Complex Analysis. Show that if $a>0$, then $\int_0^{\infty} \frac{\log(x)dx}{x^2+a^2}=\frac{\pi \log(a)}{2a}.$ The hint is ...
28 views

### Evaluation of complex integral

$$\int_{\text{c}}\frac{\sin \pi z^2 + \cos \pi z^2}{(z+1)(z+2)}$$ Where $\text{C}$ is the circle $|z| =3$ I'm a little confused about how to do this. Should this be done the normal way ? How do I ...
105 views

### Prove that a complex-valued entire function is identically zero.

Suppose $f$ is entire and $$\iint_\mathbb{C}|f(z)|^2dxdy < \infty$$Prove that $f\equiv 0.$ So far I have: Suppose $f$ is bounded. Then $f$ is constant by virtue of Liouville and so the ...
88 views

### Evaluating $\int_c\frac{1}{\sin\frac{1}{z}}\text{d}z$ over $C= \{z\big\vert|z|=\frac{1}{5}\}$

Evaluate $$\int\limits_{|z|=\frac{1}{5}} \frac{1}{\sin\frac{1}{z}}\text{d}z$$ My attempt: I know that this function has non isolated singularity at $0$, and simple poles at $\frac{1}{n \pi}$. ...
34 views

### Radius of convergence and the existence of antiderivative

I think I have some misunderstandings regarding some basic concepts. First, the question I'm dealing with is the following: Let $f$ be analytic in $\{z ;|z|>1 \}$, and $\int_{|z|=2}f(z)dz=0$. ...
33 views

### Complex integration f [duplicate]

Prove that $$\int_{0}^{\infty} \frac{dt}{1+t^{n}}=\frac{\pi}{n}\csc\frac{\pi}{n}$$ Please help,I don't have a clue
27 views

### How do I visualise the contour integral of a complex function? [duplicate]

I've just learnt about the contour integral of a complex function, but I'm having trouble figuring out what it is calculating visually. I understand it is somewhat analogous to the line integral for ...
61 views

### Determine this real integral with the Residue-theorem.

$$\int_{-\infty}^{\infty}{\frac{\sin x}{x^4-6x^2+10}\,\mathrm dx}$$ I get that when I evaluate the $\frac{\sin x}{x}$ one, I work with $\frac{e^{ix} - e^{-ix}}{2ix}$, I create a huge semicircle and a ...
41 views

### Complex integration $\Rightarrow$ delta-distribution?

My physics textbook states that $$\int_{-\infty}^{\infty} e^{i(p-p')x/\hbar}dx = 2\pi \hbar \,\delta(p-p')$$ Whereas $\delta(p-p')$ is the delta-distribution. I see that for $p=p'$ the integral ...
29 views

### Antiderivative of $\lvert z \rvert^2$

How does one determine (or show) that the complex function $f(z)=\lvert z \rvert^2$ does not have an antiderivative? (I'm assuming this because contour integrals along two different curves with the ...
77 views

34 views

44 views

44 views

### Contour integration with logarithms

I'm having trouble calculating the below integral to get the right answer: $$\frac{1}{2\pi i}\int_\gamma \frac{3}{z-2}\; dz$$ where $\gamma$ is parametrised by $\gamma(t)=3e^{it}, t\in [0,2\pi]$. So ...
30 views

### Recommendations for tutorials specifically devoted to real integration using contour integral techniques.

Complex analysis, and in particular contour integrals and the residue theory have proved a very powerful tool in computing a large class of real function integrals which would be quite troublesome to ...
153 views

### Need help with $\int_{-\infty}^\infty \frac{x^2 \, dx}{x^4+2a^2x^2+b^4}$

I'm having trouble trying to evaluate this definite integral. Mathematica didn't help much. $$\int_{-\infty}^\infty \frac{x^2 \, dx}{x^4+2a^2x^2+b^4}$$ where $a$, $b$ $\in \Bbb R^+$. Is it possible ...
58 views

18 views

### How to find integrals of the form $\int_{-r}^{+r}\frac{\gamma^{2}-1-y^{2}}{y-z\gamma} dy$

I'm trying to solve the above integral which will give a dispersion relation. Note: $\gamma$ and y are real but z is a complex variable.
25 views

### Integral problem with branch point from Physics

The question come from a Summation like this $${ \sum _{ { z=i\omega }_{ n } } { \frac { -\alpha E\pi }{ 4{ z }^{ 3 }\sqrt { -\alpha -z } } } }$$ I can use Cauchy theorem to transform it to a ...
48 views

### Closed contour integral: $\int_{\mathbb{c}}\frac{ z}{2z^{2}+1} dz$ where the contour is the unit circle

first and foremost please excuse my English. given $∫_c \frac{{z}}{2z^{2}+1}dz$ where the contour is the unit circle. so c = $e^{it}$ from 0 to $2\pi$. since the contour is the unit circle we can ...
159 views

### The complex function $\log(1+e^{iz})$

G.H. Hardy states the following: The function of the complex variable $z$ $$e^{i p z} \log(1 \pm e^{iz})\frac{1}{z^{2} \pm \theta^{2}} = f(z), \ (-1 < p <1, \theta >0),$$ ...
Let $\phi:\mathbb{C}\backslash\{\pm i,\pm 2i\}\rightarrow\mathbb{C}$ with $\phi(z)=\frac{e^{iz}}{(z^2+1)^2(z^2+4)}$. How can I find $$\int_{-\infty}^\infty\frac{\cos x}{(x^2+1)^2(x^2+4)}dx$$ ...
### Prove the integral of $Re(z)$ around a simple closed curve is imaginary
If $θ$ is any simple closed curve in $\mathbb{C}$, prove that $\int_{θ}Re(z)dz$ is pure imaginary I have shown if $θ$ is the unit circle then the answer is $\frac{i}{2}$ but I'm struggling with a ...