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 […]

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.

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 […]

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 $-R$ […]

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 […]

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 […]

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 […]

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 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.

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 […]

Intereting Posts

Definition of definition
Why are all the interesting constants so small?
Is there a common notion of $\mathbb{R}^n$, for non-integer $n$?
Where to start learning Linear Algebra?
What's the associated matrix of this linear operator?
If an IVP does not enjoy uniqueness, then there are infinitely many solutions.
What is so interesting about the zeroes of the Riemann $\zeta$ function?
Bulgarian Solitaire: Size of root loops
Polynomial $p(a) = 1$, why does it have at most 2 integer roots?
Does the Mandelbrot fractal contain countably or uncountably many copies of itself?
How to find all positive integer solutions of a Diophantine equation?
A ‘strong’ form of the Fundamental Theorem of Algebra
Volume of irregular solid
Is there a symbol to mean 'this is undefined'?
Units of a log of a physical quantity