Tagged Questions
1
vote
0answers
14 views
simplification of a complex expression
I am collecting proofs for certain integrals. To simplify certain proofs, I use
$e^{Ax}cos(Bx)=\mathcal{Re}[e^{(A+iB)x}]$, where $A$, $B$, and $x$ are real. Is there an analogous simplification for $ ...
3
votes
1answer
90 views
Integrating $\sin^2(x)$ using imaginary numbers.
I know I can change "$\sin^2\theta$" to "$\frac{1}{2}(1-\cos(2\theta))$" or use integration by parts, but I was curious about doing it using imaginary numbers. I tried this but it didn't work.
$$\int ...
4
votes
3answers
147 views
Calculate the $\int_0^{2\pi}\cos(mx)\cos(nx)dx$
I'm having trouble with this problem:
Consider the integral:
$$\tag 1\int_0^{2\pi}\cos(mx)\cos(nx)dx$$
a. Write $\cos(mx)$ and $\cos(nx)$ in terms of complex exponentials and compute ...
2
votes
1answer
96 views
Algebraic integral involving complex numbers
I need some help analytically proving the following with elementary tools:
$$\int_1^{+\infty} \frac{z^i + z^{-i}}{z^2 + 1} ~ \mathrm{d} z = \frac{\pi}{2} \mathrm{sech} \left ( \frac{\pi}{2} \right ...
4
votes
1answer
111 views
About problem in complex integrals
I solved this problem in complex integrals.
Is my answer a correct ?
Here $z$ is a complex value:
$$
C:|z-1|=1 \ \ \ \ \ \mbox{integral path}
$$
$$
\int_C\ \frac{2z^2-5z+1}{z-1}\ dz
$$
My answer
...
0
votes
2answers
142 views
How do we find the integral of this conjugate? [closed]
I'm not sure how to find
$$\int_C \overline{z}^2\ dz$$
where $C$ is the circle $|z|=2$ traveled counterclockwise once.
4
votes
2answers
102 views
Proof $\int\Re(f(x))\,\mathrm{d}x=\Re(\int f(x)\,\mathrm{d}x)$
I have a function $f: \mathbb{R}\to\mathbb{C}$. How can I proof/argue that $$\int\Re(f(x))\,\mathrm{d}x=\Re\left(\int f(x)\,\mathrm{d}x\right)$$ (and the same for the imaginary part)? I'm afraid I ...
3
votes
1answer
182 views
How do you integrate $\cos(x^n)$, specifically for $n=-1$?
How does one integrate $\cos(x^{-1})$?
I understand that the function is not defined at zero, but it is well defined, continuous, and real over the rest of $\mathbb{R}$. Nonetheless, when I put ...
2
votes
0answers
72 views
Evaluating the limit $y \to 0^+$
Given $t \in \mathbb{R}$ and $z = x + iy$ and $y>0$.
$\lim_{y\to0^+} \frac{1}{t - z} = \frac{1}{t-x} + \pi i \delta(t-x)$
This limit is given in the book Integral Transforms and Their ...
