1
vote
1answer
61 views

“Geometric” proof of Rouche's theorem on the number of zeros?

I understand the analytic proof of Rouche's theorem as presented in Stein and Shakarchi's complex analysis - $|f(z)| > |g(z)|$ on the boundary circle C ensures that the argument principle can be ...
1
vote
2answers
87 views

Intuition or figure for Reverse Triangle Inequality $||\mathbf{a}| − |\mathbf{b}|| ≤ |\mathbf{a} − \mathbf{b}|$ (Abbott p 11 q1.2.5)

I acquiesce to Wikipedia's picture for Triangle Inequality. But without referring to Triangle Inequality at all, is there intuition or figure please for Reverse Triangle Inequality for all ...
1
vote
1answer
57 views

Mentally visualizing functions of complex numbers

I've recently been learning about functions of complex numbers (to complex numbers), and I can't quite fit them into my head. When I think about real functions, I tend to mentally visualize them as ...
3
votes
2answers
185 views

What insight is supposed to be gained from this complex analysis exercise?

Let $C_0$ denote the circle centered around some point $z_0\in\mathbb{C}$ with radius $R$. We can parametrize this circle like this: $$\begin{array}{cc} z(\theta)=z_0+Re^{i\theta}, & \theta \in ...
0
votes
2answers
56 views

Why are there several roots of complex equations

I'm trying to understand why there are $n$ amount of roots in an equation of the form $z^n=$ complex equations. I understand why there are several answers to a $ \sin(x)=$ equation but I can't wrap ...
4
votes
2answers
89 views

Intuition behind the residue at infinity [duplicate]

The residue at infinity is given by: $$\underset{z_0=\infty}{\operatorname{Res}}f(z)=\frac{1}{2\pi i}\int_{C_0} f(z)dz$$ Where $f$ is an analytic function except at finite number of singular points ...
11
votes
1answer
231 views

“Why” is $[\mathbb{C}:\mathbb{R}] < \infty$?

Obviously this question is a little open-ended. A lot of complex analysis seems to work primarily because we can view $\mathbb{C}$ as a finite-dimensional $\mathbb{R}$-algebra, and apply analytic and ...
1
vote
3answers
141 views

Cauchy integral formula for derivatives intuition

A nice way to remember a formula is to connect it to something you already know. For example, to remember the Cauchy integral formula, I remember that $f(z_0)=\frac{1}{2\pi ...
2
votes
4answers
249 views

What is the Riemann Sphere?

Reading from wikipedia I understood that Riemann Sphere is used to represent extended complex plane. But it would be great if someone could explain it in a less technical manner.
1
vote
2answers
112 views

Uniform convergence of $\sum_{n=0}^{\infty} \frac{(-1)^n 2i}{(2n+1)z^{2n+1}}$ on squarewith vertices $\pm6\pm6i$?

Can someone explain me how I can check whether the convergence of $\sum_{n=0}^{\infty} \frac{(-1)^n 2i}{(2n+1)z^{2n+1}}$ is uniform on the (boundary of) square $A$ with vertices $\pm6\pm6i$?
1
vote
1answer
163 views

How to recognize removable singularity and how to remove it

I don't understand the idea of a removable singularity yet. Can someone explain me how to recognize a removable singularity and how to remove it? Example: $g(z)=f(z)/z$. Is $z=0$ then a removable ...
1
vote
0answers
37 views

Intuitive understanding of Lemma of Jordan

Can somebody give me an intuitive understanding of the Lemma of Jordan, which is: $$\lim \limits_{R\rightarrow \infty} \int_{\gamma} \exp(i \omega z)\,\rm dz=0 $$ if: $$\lim \limits_{z\rightarrow ...
1
vote
0answers
108 views

Finding the derivative of a piecewise function of a complex variable

I am working on an assignment that involves finding the derivative evaluated at zero of this piecewise function of a complex variable: Let $g:\mathbb{C}\rightarrow\mathbb{C}$ be defined by: ...
6
votes
1answer
454 views

What is contour integration

So I recently took a course that involves contour integration. I understood how to perform the integrals out, but I never got a hold of what the physical meaning was. I understand the introductory ...
2
votes
2answers
185 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} = ...
1
vote
1answer
53 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
252 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 ...
9
votes
1answer
844 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
302 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
243 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
487 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} ...
18
votes
4answers
865 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
141 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 ...
6
votes
2answers
1k 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
354 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
802 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
560 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
252 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 ...
13
votes
3answers
313 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
290 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) = ...
48
votes
12answers
4k 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 ...