Tagged Questions
0
votes
2answers
64 views
Whats the connection between functions with curl 0 and holomorphic functions
When I first saw the Cauchy-Riemann differential equations they remind me on the conditions for the curl of a function to be zero.
Here some notation I will use:
$$\frac{\partial f}{\partial x} = ...
0
votes
1answer
31 views
Intuition Behind Krantz Theorem
The theorem I'm referring to is as follows:
Let $z_0$ be a root of a nonzero holomorphic function $f$ , and let $n$ be the least positive integer such that, the $n$-th derivative of $f$ evaluated ...
2
votes
1answer
81 views
Can someone please explain Morera's Theorem and the Schwarz Reflection Principle?
I have been reading Complex Analysis, Third Edition by Joseph Bak and Donald J. Newman and am stuck with chapter 7 which deals with the mentioned theorem. I have read through the chapter at least ...
6
votes
1answer
158 views
What is the intuition behind the Wirtinger derivatives?
The Wirtinger differential operators are introduced in complex analysis to simplify differentiation in complex variables. Most textbooks introduce them as if it were a natural thing to do. However, I ...
1
vote
1answer
156 views
Intuitive explanation of Residue theorem in Complex Analysis
The residue theorem that states that if
a) $U$ is a simply connected and open subset of the complex plane,
b) $a_1,\dots,a_n$ are finitely many points of $U$,
c) and $f$ is a function which is ...
6
votes
2answers
166 views
How to 'analyze' problems in analysis; Computing $\int_0^{2\pi}\frac{1}{(a+b\cos(\theta))^2}d\theta$
If $a, b \in \mathbb{R}$ with $a > b > 0$, compute this ungodly thing;
$$\int_0^{2\pi}\frac{1}{(a+b\cos(\theta))^2}d\theta$$
I'm really not a fan of complex analysis... I can't visualize ...
7
votes
1answer
315 views
Physical interpretation of the generating function for the Bessel functions.
It is well known that the generating function for the Bessel function is
$$f(z) = \exp \left (\frac12 \left (z - \frac1z \right ) w \right ).$$
So, we have
$$f(z) = \sum_{\nu = -\infty}^{\infty} ...
13
votes
3answers
473 views
Perspectives on Riemann Surfaces
So, I have come to a somewhat impasse concerning my class selection for next term, and I have exhausted all the 'biased' sources. So, I was wondering if anyone in this fantastic mathematical community ...
0
votes
2answers
126 views
Identification of complex plane as $R^2$.
If we have following identification:
$$(x,y)\to (z,\overline{z})$$
We will have $$\frac{\partial}{\partial x}= \frac{\partial}{\partial z}+\frac{\partial}{\partial \overline{z}}$$
and ...
5
votes
2answers
486 views
Line integration in complex analysis
In normal line integration, from what I understand, you are measuring the area underneath $f(x,y)$ along a curve in the $x\text{-}y$ plane from point $a$ to point $b$.
But what is being measured with ...
5
votes
1answer
279 views
Arnold's Trivium problem 52
Calculate the first term of the asymptotic expression as $k \to \infty$ of the integral
$$ \int_{-\infty}^{+\infty}\frac{e^{ikx}}{\sqrt{1+x^{2n}}}dx $$
May I bother you to explain what the ...
5
votes
2answers
469 views
Intuition behind euler's formula [duplicate]
Possible Duplicate:
How to prove Euler's formula: $\\exp(i t)=\\cos(t)+i\\sin(t)$ ?
Hi, I've been curious for quite a long time whether it is actually possible to have an intuitive ...
7
votes
1answer
416 views
Intuition Building: Visualizing Complex Roots
I can graph the parabola $y = x^2 + 1$. I see that it does not intersect the $x$-axis, and therefore it must have complex roots, namely $+i$ and $-i$. I can plot these roots on an Argand diagram at ...
0
votes
1answer
203 views
Complex integrals and the possibility of avoiding parametric equations
I've been playing around with this equation:
$\displaystyle\int_{-\pi}^\pi{\displaystyle\frac{1-e^{3it}}{1-e^{it}}dt}$
Now it seems to me that we can (possibly) split the integral into four ...
12
votes
2answers
250 views
What can be gleaned from looking at a domain-colored graph of a complex function?
Functions from $\mathbb{C} \rightarrow \mathbb{C}$ are hard to visualize because of their 4-dimensional nature. One nice way of looking at them is by what's called domain coloring. An example from the ...
1
vote
0answers
275 views
How effective is this alternative to integration?
I have a function that is difficult to integrate. So I elect to work with power series representations. Suppose the power series representation for this function is the following:
$f(x) = ...
46
votes
12answers
3k views
Intuitive explanation of Cauchy's Integral Formula in Complex Analysis
There is a theorem that states that if $f$ is analytic in a domain $D$, and the closed disc {$ z:|z-\alpha|\leq r$} contained in $D$, and $C$ denotes the disc's boundary followed in the positive ...
