Articles of residue calculus

Integrating $\frac{1}{1+z^3}$ over a wedge to compute $\int_0^\infty \frac{dx}{1+x^3}$.

Compute $\displaystyle\int_0^\infty \frac{dx}{1+x^3}$ by integrating $\dfrac{1}{1+z^3}$ over the contour $\gamma$ (defined below) and letting $R\rightarrow \infty$. The contour is $\gamma=\gamma_1+\gamma_2+\gamma_3$ where $\gamma_1(t)=t$ for $0\leq t \leq R$, $\gamma_2(t)=Re^{i\frac{2\pi}{3}t}$ for $0\leq t \leq 1$, and $\gamma_3(t)=(1-t)Re^{i\frac{2\pi}{3}}$ for $0\leq t \leq 1$. So, the contour is a wedge, and by letting $R\rightarrow \infty$ we’re integrating over one […]

Residue at infinity (complex analysis)

I have trouble with the residue : at $z = \infty$. I tried to solve it at $z=0$ but it turns out that I was wrong while $z=0$ is not a pole. I must solve it at $z=2$ but I’m stuck. Any suggestion will be much appreciated.

Integration using residues

For the following problem from Brown and Churchill’s Complex Variables, 8ed., section 84 Show that $$ \int_0^\infty\frac{\cos(ax) – \cos(bx)}{x^2} \mathrm{d}x= \frac{\pi}{2}(b-a)$$ where $a$ and $b$ are positive, non-zero constants, by integrating about a suitable indented contour. The contour in question is the upper half of an annulus bisected by the $x$-axis with an outer radius […]

How to do contour integral on a REAL function?

Suppose we are given the problem: Evaluate: $$\int_{0}^{\infty} \frac{1}{x^6 + 1} dx$$ Where $x$ is a real variable. A real variable function (no complex variables). I was reading Schaum’s outline to this solution and it defines: $f(z) = \displaystyle \frac{1}{z^6 +1}$ And it says consider a closed contour $C$ Consisting of the line from $-R$ […]

Singular points

Find all the (isolated) singular points of $f$, classify them, and find the residue of $f$ at each singular point. $$f(z) = \frac{z^{1/2}}{z^2 + 1}$$ I think I have $3$ singularities at $z=0,-i$ and $i$ but am unsure about what to do next and what type of singularities they are. I don’t fully understand singularities […]

How to calculate this Ei(x)-involved definite integral?

I want to solve the integral attached below by means of residue theorem. I tried the common integration ways and seeked references(e.g, Rjadov, et. al). Finally, I decided to solve this integral by means of ” Residue Theorem “. Can anyone help me ? $$ \int_u^\infty \frac{Ei(-x)e^{-px}}{(x-\beta)}dx\,. $$ $$ p>0\ ,\ \beta>0\ \ ,\ u>0 […]

Residue at essential singularity

I need a little help with the following problem. I’ve tried many ways, but i didnt succeed. I think there needs to be a trick or something, some transformation. The task is to find the residue of the function at its singularity e.g. z=-3 \begin{equation} f(z)=\cos\left(\frac{z^2+4z-1}{z+3}\right) \end{equation} I tried to write it as \begin{align} \cos\left(\frac{z^2+4z-1}{z+3}\right)=1-\frac{1}{2!}\left((z+1)-\frac{4}{z+3}\right)^2+\frac{1}{4!}\left((z+1)-\frac{4}{z+3}\right)^4-\frac{1}{6!}\left((z+1)-\frac{4}{z+3}\right)^6+\ldots […]

$\int_0^{2\pi}e^{a \cos{\theta}}\cos({\sin{\theta}})\,d\theta$ using residues

How do I find the following integral by converting it into a complex integral and then using residue theorem? $$\int_0^{2\pi}e^{a \cos{\theta}}\cos({\sin{\theta}})\,d\theta$$ My approach is as follows. Substitute $z=e^{i\theta}$ so that $\cos{\theta}=\frac{z+z^{-1}}{2}$, and similarly for sine. For the cos outside the exponent, use this substitution again. But it is just giving me a string of exponents, […]

Evaluate $\int_0^{\pi} \frac{\sin^2 \theta}{(1-2a\cos\theta+a^2)(1-2b\cos\theta+b^2)}\mathrm{d\theta}, \space 0<a<b<1$

Evaluate by complex methods $$\int_0^{\pi} \frac{\sin^2 \theta}{(1-2a\cos\theta+a^2)(1-2b\cos\theta+b^2)}\mathrm{d\theta}, \space 0<a<b<1$$ Sis.

Calculate using residues $\int_0^\infty\int_0^\infty{\cos\frac{\pi}2\Big(nx^2-\frac{y^2}n\Big)\cos\pi xy\over\cosh\pi x\cosh\pi y}dxdy,n\in\mathbb{N}$

Q: Is it possible to calculate the integral $$ \int\limits_0^\infty \int\limits_0^\infty\frac{\cos\frac{\pi}2 \left(nx^2-\frac{y^2}n\right)\cos \pi xy}{\cosh \pi x\cosh \pi y}dxdy,~n\in\mathbb{N}\tag{1} $$ using residue theory? For example, when $n=3$ $$ \int\limits_0^\infty \int\limits_0^\infty\frac{\cos\frac{\pi}{2} \left(3x^2-\frac{y^2}{3}\right)\cos \pi xy}{\cosh \pi x\cosh \pi y}dxdy=\frac{\sqrt{3}-1}{2\sqrt{6}}. $$ There is a closed form formula to calculate (1) for arbitrary natural $n$, but I don’t know how […]