Articles of contour integration

How to show the contour integral goes to $0$ of semicircle?

Consider the integral: $$\int_{0}^{\infty} \frac{\log^2(x)}{x^2 + 1} dx$$ Image taken and modified from: Complex Analysis Solution (Please Read for background information). $R$ is the big radius, $\delta$ is the small radius. We consider $\displaystyle f(z) = \frac{\log^2(z)}{z^2 + 1}$ where $z = x+ iy$ How can we prove: $$\oint_{\Gamma} f(z) dz \to 0 \space \text{when} […]

Evaluating an integral across contours: $\int_C\text{Re}\;z\,dz\,\text{ from }-4\text{ to } 4$

This is for an assignment, describing the procedure is most beneficial for me, rather that solely computing the result. I want to evaluate the following integral: $$\int_C\text{Re}\;z\,dz\,\text{ from }-4\text{ to } 4$$ Along the line segments from $-4$ to $-4-4i$ to $4-4i$ to $4$, e.g: So I want to evaluate three integrals of the above […]

Change the order of integrals:$\int_0^1dx\int_0^{1-x}dy\int_0^{x+y}f(x,y,z)dz$

$$\int_0^1dx\int_0^{1-x}dy\int_0^{x+y}f(x,y,z)dz$$ From this it is obvious that $x\in[0,1],y\in[0,1-x],z\in[0,x+y]$. For it asks for the order to be in $$\int dz\int dx\int f(x,y,z)dy$$ . My method of doing this is for start from the outer variable ,in this case z, and from the original one can deduct that $z\in [0,1]$ But the professor goes on to make […]

Closed form of $\int_0^\infty \frac{dt}{\left(i t\sqrt{3}+\ln(2\sinh t)\right)^2}$

While evaluating the integral $$ I_1=\int_{0}^\infty\frac{\sin\pi x~dx}{x\prod\limits_{k=1}^\infty\left(1-\frac{x^3}{k^3}\right)}, $$ I came to this integral of elementary function $$ I_2=\int_0^\infty \frac{dt}{\left(i t\sqrt{3}+\ln(2\sinh t)\right)^2}. $$ In fact $I_2$ is real and $$ I_1=-2\pi I_2. $$ Brief outline of proof is as follows. Write the infinite product in terms of Gamma functions, apply reflection formula for Gamma function to […]

Residue of complex function

I know that if function is meromorphic then it will have resiue. My simple question is if function is not meromorphic then can it have a residue, because I am not getting any such statement in my book. If it is correct or incorrect then plz help me with suitable example.

Evaluating a trigonometric integral by means of contour $\int_0^{\pi} \frac{\cos(4\theta)}{1+\cos^2(\theta)} d\theta$

I am studying for a qualifying exam, and this contour integral is getting pretty messy: $\displaystyle I = \int_0^{\pi} \dfrac{\cos(4\theta)}{1+\cos^2(\theta)} d\theta $ I first notice that the integrand is an even function hence $\displaystyle I = \dfrac{1}{2} \int_{-\pi}^{\pi} \dfrac{\cos(4\theta)}{1+\cos^2(\theta)} d\theta $ Then make the substitutions $\cos(n\theta) = \dfrac{e^{in\theta}+e^{-in\theta}}{2}$, and $z=e^{i\theta}$ to obtain: $\displaystyle I = […]

Evaluate the Bessel Function $J = \int^{2\pi}_{0}{e^{\cos x}}{\cos(2x – \sin x)}\, dx$

I need to evaluate the following definite integral: $$J = \int^{2\pi}_{0}{e^{\cos x}}{\cos(2x – \sin x)}\, dx$$ I have attempted basic variable substitution and expanding the cosine term, but I have not been able to find an indefinite integral. I believe the best strategy would be to use contour integration, but I am not sure on […]

Contour Integration – my solution for real integral is complex?

So I’ve had a crack at this contour integration question and have somehow managed to get a complex solution for a real integral… I’ve gone through my working a number of times but can’t seem to find the mistake, so was hoping someone here could help. Evaluate the integral $$ I=\int^\pi_{-\pi} \frac{\,d\theta}{a+b\cos\theta+c\sin\theta} $$ where $a$, […]

Integral $\int_0^{2\pi}\frac{dx}{2+\cos{x}}$

How do I integrate this? $$\int_0^{2\pi}\frac{dx}{2+\cos{x}}, x\in\mathbb{R}$$ I know the substitution method from real analysis, $t=\tan{\frac{x}{2}}$, but since this problem is in a set of problems about complex integration, I thought there must be another (easier?) way. I tried computing the poles in the complex plane and got $$\text{Re}(z_0)=\pi+2\pi k, k\in\mathbb{Z}; \text{Im}(z_0)=-\log (2\pm\sqrt{3})$$ but what […]

Evaluation of $\int_0^\infty \frac{x^2}{1+x^5} \mathrm{d} x$ by contour integration

Consider the following integral: $$\int_{0}^{\infty} \frac{x^{2}}{1+x^{5}} \mathrm{d} x \>.$$ I did the following: Since $-1$ is a pole on the real axis, I took $z_{1}=e^{3\pi/5}$ then constructed an arc between $Rz_{1}$ and $R$ : $f(e^{2\pi i /5}z)=f(z)$ it follows that : $(1-e^{2pi i/5})\int_{\alpha}f(z)dz + \int_{\beta}f(z)dz = 2\pi i \text{ Res } z_{2} f $ , […]