What is Complex variables: Definition and 120 Discussions
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics; as well as in physics, including the branches of hydrodynamics, thermodynamics, and particularly quantum mechanics. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering.As a differentiable function of a complex variable is equal to its Taylor series (that is, it is analytic), complex analysis is particularly concerned with analytic functions of a complex variable (that is, holomorphic functions).
I tried saying z = x + iy, then squared both sides so that I would get something that looked like:
|z - i|^2 + |z + i|^2 + |z - i||z + i| = 3, where the first two terms are simple but the third term is what I don't know what to do with. I'm wondering if I'm using the wrong approach.
For that...
Homework Statement
Homework Equations
The relevant equation is that sqrt(z) = e^(1/2 log z) and the principal branch is from (-pi, pi]
The Attempt at a Solution
The solution is provided, since this isn't a homework problem (I was told to post it here anyway). I don't understand why the...
(mentor note: this is a homework problem with a solution that the OP would like to understand better)
In Taylor's Complex Variables,
Example 1.4.10
Can someone help me understand this? I don't know what they mean by (i, i inf), or how they got it and -it
I'm learning complex analysis right now, and I'm reading from Joseph Taylor's Complex Variables.
On Theorem 1.4.8, it says "If a log is the branch of the log function determined by an interval I, then log agrees with the ordinary natural log function on the positive real numbers if and only if...
Homework Statement
Determine if the following function is continuous: f(x) = (x-iy)/(x-1)
Homework Equations
How do find out if a function is continuous without graphing it and without a point to examine? I know I've learned this, probably in pre-calculus too, but I'm blanking
The Attempt at...
Hello, I am a rising sophomore in Astronomy and Physics. I am taking complex variables next semester and was wondering the effort required to succeed in the class. There are some other classes I'd like to take, however I don't want to overload myself. I have taken up through multivariable calc...
Homework Statement
I have never formally studied complex analysis, but I am reading this paper: http://adsabs.harvard.edu/abs/1996MNRAS.283..837S
wherein section 2.2 they make use of the residue theorem. I am trying to follow along with this (and have looked up contour integration, cauchy's...
Suppose that f is analytic on the disc $\vert{z}\vert<1$ and satisfies $\vert{f(z)}\vert\le{M}$ if $\vert{z}\vert<1$. If $f(\alpha)=0$ for some $\alpha, \vert{\alpha}\vert<1$. Show that,
$$\vert{f(z)}\vert\le{M\vert{\frac{z-\alpha}{1-\overline{\alpha}z}}\vert}$$
What I have:
Let...
Suppose the polynomial p has all its zeros in the closed half-plane $Re w\le0$, and any zeros that lie on the imaginary axis are of order one.
$$p(z)=det(zI-A),$$
where I is the n x n identity matrix.
Show that any solution of the system
$$\dot{x}=Ax+b$$
remains bounded as $t\to{\infty}$...
We define the Legendre polynomial $P_n$ by
$$P_n (z)=\frac{1}{2^nn!}\frac{d^n}{dz^n}(z^2-1)^n$$
Let $\omega$ be a smooth simple closed curve around z. Show that
$$P_n (z)=\frac{1}{2i\pi}\frac{1}{2^n}\int_\omega\frac{(w^2-1)^n}{(w-z)^{n+1}}dw$$
What I have:
We know $(w^2-1)^n$ is analytic on...
Studying for my complex analysis final. I think this should be a simple question but wanted some clarification.
"Extend the formula
$$\frac{1}{2i\pi} \int_\omega \frac{h'(z)}{h(z)}\, dz = \sum_{j=1}^N n_j - \sum_{k=1}^M m_k$$
to prove the following.
Let $g$ be analytic on a domain...
<Moderator's note: moved from a technical forum, so homework template missing>
Hi. I have solved the others but I am really struggling on 22c. I need it to converge for |z|>2. This is the part I am really struggling with. I am trying to get both fractions into a geometric series with...
I'm currently an applied math major. I'm creating a schedule for my next semester and I have the choice to take either complex variables or vector analysis with linear algebra and a college geometry course(elective of choice), but I don't know which pairing will be less stressful. I am currently...
Homework Statement
evaluate ##\int \frac{sinh(ax)}{sinh(\pi x)}## where the integral runs from 0 to infinity
Homework Equations
The Attempt at a Solution
consider ##\frac{sinh(az)}{sinh(\pi z)}##
Poles are at ##z= n \pi i##
So I'm considering the contour integral around the closed contour...
I found this formula in a paper:
\int exp( \frac{x1 + i x2}{ \sqrt 2} \eta^* - \frac{x1 - i x2}{ \sqrt 2}
\eta) D(\eta)/ \pi d^2 \eta
the author calls it the Fourier transform of D.
It is the first time thar i see this formula.
How common is this notation? Can we use it without problem?
Dear all,
I'd like to know what is the place/use of complex variables (and complex analysis) in classical mechanics. By the way, is there any?
Thanks for your help. Best regards!
Today, I had a class on Complex analysis and my professor wrote this on the board :
The Laplacian satisfies this equation :
where,
So, how did he arrive at that equation?
I am trying to solve a Duffing's equation ##\ddot{x}(t)+\alpha x(t)+\beta x^3(t)=0## where ##\alpha## is a complex number with ##Re \alpha<0## and ##\beta>0##. The solution can be written as Jacobi elliptic function ##cn(\omega t,k)##. Then both ##\omega## and ##k## are complex. The solution to...
Homework Statement
Find the principle argument Arg z when
z = (sqrt(3) - i)^6
Homework Equations
The Attempt at a Solution
I'm sorry to say that I'm not sure how to solve this problem. It's my understanding that what this question is basically asking me to do is find theta such that...
Homework Statement
Describe the set of points determined by the given condition in the complex plane:
|z - 1 + i| = 1
Homework Equations
|z| = sqrt(x2 + y2)
z = x + iy
The Attempt at a Solution
Tried to put absolute values on every thing by the Triangle inequality
|z| - |1| + |i| = |1|...
Some calculators say (-2)2/3 is equal to ##-\frac{1}{2^\frac{1}{3}}+i\frac{3^\frac{1}{2}}{2^\frac{1}{3}}## while others say its equal to ##4^{\frac{1}{3}}## i.e. ##|-\frac{1}{2^\frac{1}{3}}+i\frac{3^\frac{1}{2}}{2^\frac{1}{3}}|##.
I think I am right to imply from above that (-2)2/3 does have an...
I have seen in the online Stanford Encyclopedia of Philosophy in the entry on Copenhagen Interpretation of Quantum Mechanics that Niels Bohr had argued that the theory of relativity is not a literal representation of the universe:
"Neither does the theory of relativity, Bohr argued, provide us...
Is Complex Analysis and Complex Variables the same thing? Is Complex Analysis pure or applied math? Is Complex Variables pure or applied math? What's the prerequisite of Complex Analysis and Complex Variables? Are they useful for the field of computer science?
Homework Statement
Find the Laurent series expansion of f(z) = \log\left(1+\frac{1}{z-1}\right) in powers of \left(z-1\right).
Homework Equations
The function has a singularity at z = 1, and the nearest other singularity is at z = 0 (where the Log function diverges). So in theory there should...
Homework Statement
Use the power series for e^z and the def. of sin(z) to check that
sum ((-1)^k z^(2 k+1))/((2 k+1)!)
Homework Equations
The Attempt at a Solution
I apologize, but I am not particularly good with latex. Therefore, I attached a picture of my solution thus far...
Homework Statement
I hate to upload the whole problem, but I am trying to evaluate an indefinite integral, and I can follow the solution until right near the end. The example says that for a point on C_R|e^{-3z}|=e^{-3y}\leq 1. I don't understand how they can say this. Below is the question...
Homework Statement
Obtain the Taylor series ez=e Ʃ(z-1)n/n! for 0\leq(n)<\infty, (|z-1|<\infty) for the function f(z)=ez by (ii) writing ez=ez-1e.
Homework Equations
Taylor series:
f(z) = Ʃ(1/2\pi/i ∫(f(z)/(z-z0)n+1dz)(z-z0)n
The Attempt at a Solution
The first part of this...
Homework Statement
Show that if C is a positively oriented simple closed contour, then the area of the region enclosed by C can be written (1/2i)/∫C\bar{}zdz.
Note that expression 4 Sec. 46 can be used here even though the function f(z)=\bar{}z is not analytic anywhere.
FORMATTING NOTE: SHOULD...
Hello, I was looking at Riley's solution manual for this specific problem. Along the way, he ended up with a quadratic inequality:
If you looked at the image, he said it is given that λ is real, but he asserted that λ has no real roots because of the inequality. Doesn't that mean λ is...
This is the statement, in case you're not familiar with it.
Let ## f_j(w,x), \; j=1, \ldots, m ## be analytic functions of ## (w,z) = (w_1, \ldots, w_m,z_1,\ldots,z_n) ## in a neighborhood of ##w^0,z^0## in ##\mathbb{C}^m \times \mathbb{C}^n ## and assume that ##f_j(w^0,z^0)=0, \...
As an electrical engineering student can learning about complex variables be beneficial to me? If so can someone recommend an introductory book that I can read on my own? Also I have very little experience working with complex variables.
ℂI am working on an assignment and have come across a question that I'm not quite sure how to approach. Here it is, with my "solution" and reasoning:
"[F]ind the limit at ∞ of the given function, or explain why it does not exist.
24. h(z) = Arg z , z \neq 0" (Complex Variables Second...
Author: James Brown, Ruel Churchill
Title: Complex Variables and Applications
Amazon link https://www.amazon.com/dp/0073051942/?tag=pfamazon01-20
Table of Contents:
Preface
Complex Numbers
Sums and Products
Basic Algebraic Properties
Further Properties
Vectors and Moduli
Complex...
I know UC San Diego is good, Rothschild and Ebenfelt are there, but are there any other ones that stand out in these and related fields?
Thanks a bunch
Homework Statement
Determine the set on which f(z) = 1/(z^3 +1) is analytic and compute its derivative.
Homework Equations
Hint: you do not need to appeal to the Cauchy-Riemann equations
The Attempt at a Solution
Total stuck with this one. everything we have done this far has...
Hi everyone,
I'm a Physics student going into my Junior year and I'm currently registering for my courses for the following semester and I have two options for my "complex" course, namely:
---------------------------------------------------
Complex Variables
Theory of functions of one complex...
Homework Statement
The Fortran 90 code shown below was written to solve a Crank-Nicholson algorithm that describes the motion of a quantum particle (hence the need for complex numbers). Probably not that crucial to know the details of the math/physics, just the code...
The problem is that...
I'm not sure if it's OK to post this question here or not, the Calculus and Beyond section doesn't really look very heavily proof oriented.
I'm trying to prove that if continuous complex valued function f(z) is such that the directional derivatives(using numbers with unit length) preserve...
So I'm taking my complex variables class and learning about these cool powerful theorems like the Cauchy Goursat theorem. I know this all has huge application in physics however I just don't know what they are. Currently I'm only taking freshmen E@M so I know I won't be using it there. But next...
Homework Statement
a) \lim_{z\to 3i}\frac{z^2 + 9}{z - 3i}
b) \lim_{z\to i}\frac{z^2 + i}{z^4 - 1}
Homework Equations
?
The Attempt at a Solution
I'm assuming both of these are very, very similar, but I'm not quite sure how to solve them. I would like a method other than using ε...
Homework Statement
Solve for a, a \in \mathbb{C}
\frac{2\ln(a^2 - 1)}{\pi i} = 1
Homework Equations
N/A.
The Attempt at a Solution
Reorganizing the equation.
2\log(a^2 - 1) = \pi i
We can show that
\int_{0}^\infty e^{-kx}dx=\frac{1}{k}
for real $$k>0.$$
Does this result hold for $$\Re k>0$$ belonging to complex numbers? The reason I have this question is because $$i\times\infty$$ is not $$\infty$$ and so u substitution would not work.
Complex Variables - Mappings under e^z
Homework Statement
Find the image of S1,S2 under ez.
S1 = {z=x+iy : 0 < y < \pi }
S2 = {z=x+iy : x > 0, 0 < y < \pi }
Homework Equations
w=ez
w=\rhoei\varphi
\rho=ex, \varphi=y
The Attempt at a Solution
Did not know how to get started. I...
Homework Statement
Find all solutions to (z2+1)2=-1
The Attempt at a Solution
I know that because it is a polynomial of degree 4 it is a square inscribed inside of a circle in the complex plane. All i really need is one solution and from that finding the other three is easy. I have tried...
When dealing with electrodynamics it is usual to use complex variables for the electromagnetic field while taking into account that the electromagnetic field is real and that at the end one has to take the real part of the complex solution for the field. However, what happens to compound...
Homework Statement
For α > 0, determine u(x) by the inverse Fourier transform
u(x) = \frac{1}{2\pi}\int_{-\infty}^{\infty}\ \frac{e^{ikx}}{ik+\alpha}\ dk
Homework Equations
The Attempt at a Solution
This seemed like a relatively simple residue problem. You just note that...
Hi, I am a math and physics major planning on going into biophysics for grad school, and i want to do computational/mathematical modelling/theoretical work in the field. I have one more math course to take and I am not sure which would be more useful. Here are their very brief course...