# Tagged Questions

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

51 views

### Integration of 1/(square root of a third order polynomial) on the complex plane

I have to compute the integral $$\oint_{a,b} \frac{d \lambda}{\sqrt{(\lambda-\lambda_1)(\lambda-\lambda_2)(\lambda-\lambda_3)}},$$ Here is the picture of the integration contours and cuts: The "...
29 views

### Integral of $\omega\wedge\overline{\omega}$ on Riemann surface

Let $X$ be a Riemann surface of genus $g$ and $\omega$ a meromorphic 1-form on it. I've read that if $\omega$ has just a simple pole in $x\in X$ (and is holomorphic on $X\setminus\{x\}$) then the ...
42 views

### Is $\int_{\gamma} \sec ^2z \ \mathrm{d}z=0$?

Let $\gamma = \gamma(0;2)$. Is $$\int_{\gamma} \sec ^2z \ \mathrm{d}z$$ equal to $0$? I'm trying to answer this question using only tools like Cauchy Theorem or the Deformation Theorem since ...
36 views

39 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 ...
53 views

34 views

25 views

99 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 ...
91 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 ...
233 views

### A variation of Ahmed's integral

Given that the closed form exist, evaluate the following Integral: $$\displaystyle \int\limits_{0}^{1} \frac{(x^2+4)\sin^{-1}x}{x^4-12x^2+16} \, dx$$
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 $\frac{1}{n \pi}$. ...
29 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 ...
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 $\int_{|z|=2}f(z)dz=0$. ...
119 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 ...
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
28 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 ...
63 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 ...
42 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 ...
83 views

44 views

### Winding number of closed curves

Let $c_1,c_2$ be closed curves in $\mathbb C^{\times}$ and we define $c(t):=\frac{c_1(t)}{c_2(t)}$. Proof the following for the winding number $win(c,0)=win(c_1,0)-win(c_2,0)$. I have no idea to ...
18 views