What is Complex analysis: Definition and 780 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).
Im a rising junior in the US starting my upper division physics classes.
I have an opening this quarter and want to take an applied math course, but cannot decide between these two:
In the mathematics department:
"Applied complex anlysis
Introduction to complex functions and their applications...
So I will be a sophomore this next semester, and I am having difficulty deciding whether or not to take complex analysis. I am majoring in chemical and biomolecular engineering (with a concentration in cellular/molecular engineering), but I feel after this past semester my heart really lies...
Homework Statement
Suppose that f is an entire function. Define g(z)=f*(z*), where * indicates conjugates. I know from another problem that g(z) is also entire. Suppose also that f(z) maps the real axis into the real axis, so that f(x+0i)is in R for at x in R. Show that f(z)=g(z) for all z in...
Homework Statement
Q. (a) State Liouville's Theorem
(b) Suppose that f is analytic in C and satisfies f(z + m + in) = f(z) for all integers m,n . Prove f is constant.
Homework Equations
The Attempt at a Solution
(a) Liouville's Theorem - If f is bounded and analytic in C, then...
As I am studying for an exam I am trying to wrap my head around the concepts I learned. I want to make sure I fully understand the concepts before the exam in 1.5 weeks.
Cauchy's Theorem
If u and v satisfy the Cauchy-Riemann equations inside and on the simple closed contour C, then the...
Homework Statement
Given four complex numbers, z1, z2, z3, and z4, show that (z4-z1)(z3-z2)/(z4-z2)(z3-z1) is real if and only if the four points lie on a circle or a line
Homework Equations
polar form of complex numbers: z=|z|e^(iarg(z))
The Attempt at a Solution
Let r be the...
Homework Statement
f(z) is a complex function (not necessarily analytic) on a domain D in C. The directional derivative is Dwf(z0)=lim(t->0) (f(z0+tw)-f(z0))/t, where w is a unit directional vector in C. There are three parts to the question:
a. Give an example of a function that is not...
Homework Statement
The problem is from Sarason, page 44, Exercise IV.14.1.
Let f be a univalent holomorphic function in the open connected set G, and let g be the inverse function.
Assume that f(G) is open, that g is continuous, and that f\prime\neq 0\forall z\in G. Prove g is...
Homework Statement
The problem, for reference, is from Sarason's book "Complex Function Theory, 2nd edition" and is on page 81, Exercise VII.5.1.
Let C be a counterclockwise oriented circle, and let f be a holomorphic function defined in an open set containing C and its interior. What is...
Homework Statement
This question is in my exam review problem from my complex analysis class.
Compute f(100)(0)/100!, where f(z) = 1/(1+i-sqrt(2)z).
(f(100)(0) means the 100th derivative of f evaluated at 0.)
Homework Equations
Cauchy's integral formula might be helpful.
The answer to this...
Being a high school student who will be going into physics, should I take complex analysis or abstract algebra in the fall? I can't take both at once, and I am set to take intro to QM (I will already have taken Calc I-III, an introductory functional analysis course, and linear algebra. I also...
Homework Statement
This seems to be just a simple limit problem and I feel like I should know it but I'm just not seeing it.
I have a continuous function f, and a fixed w
I want to show that the limit (as h goes to 0) of the absolute value of:
(1/h)*integral[ f(z)-f(w) ]dz = 0...
I'm going to be taking the graduate complex analysis this coming Fall and I've not taken the undergraduate version of the course. It will be a challenge but something that my advisers told me will be surely doable. Anyway, aside from the textbook used for the course, can anyone recommend a...
Homework Statement
What is the function (linear transformation) that maps z_{1} = 2 and z_{2} = -3i onto w_{1} = 1+i and w_{2} = 3?
I think it's asking for the function that if you put 2 in it, it should give 1+i, and if you put -3i in the same function, it should give 3.
The answer...
I am not a mathematician but I have noticed how strangley similar the treatments of curvature and residues are when you compare the residues of residue calculus and the curviture of the gauss bonet forumlation of surfaces. Is there some generalization of things that contains both of these...
Homework Statement
Let gama be a closed curve and f be analytic function. Show that the integration of f(z)f' dz is puerly imaginary
Homework Equations
The Attempt at a Solution
Hello All,
Just when I thought I understood whatever there was to understand about Normal Families...
F(z) is analytic on the punctured disk and we define the sequence
f_{n}=f(z/n) for n \leq 1.
Trying (and failing) to show that {f_n} is a normal family on the punctured disk iff the...
I'm not very clear of the problems below,so I may make some mistakes,if you point out them and explain to me,I'm reallly grateful.
1.If f(z) is an analytic function,why can we derivate it as a real function to get it's derivation?
I mean f'(z) should be f^' (z) = \frac{{\partial...
Complex analysis has a lot of nice theorems that real analysis doesn't have: if you can take the complex derivative once, you can take it \infty many times. Maximum modulus theorem; inside the radius of convergence the Taylor series of a function converges to the function.
So what I wonder is...
Homework Statement
Let f be analytic at the complex plane excapt for z= -1 and z=3 which are simple poles of f.
Let \Sigma_{-\infty}^{-1} a_{n}(z-2)^{n} be the Laurent series of f.
In part A I've found that the series converges at 1<|z-2|<3 .
B is: Find the coeefficients a_{n} of the...
Homework Statement
let g denote the elliptic arc parametrized by z(t) = 2cost + 3isint, for t between 0 and pi/2 (inclusive).
Evaluate the integral of f(z) = z[sin(pi*z^2) - cos(pi*z^2)] over g.
Homework Equations
If g is determined by the function z mapping from [a,b] to C and...
I have a degree in Engineering. Now I am back to school, for a 2 year Master's degree in Statistics. The second semester just started. And there will be a 3rd. Is there a chance that I will need complex numbers? My background in Complex Analysis is very limited. Should I study any Complex...
Homework Statement
show that the function
F:C\rightarrowC
z \rightarrow z+|z|
is continuous for every z0\in C2. Proof
F is continuous at every z0\in C if given an \epsilon > 0 , there exists a \delta > 0 such that \forall z 0 \in C, |z-z 0|< \delta implies |F(z)-F(z0)|< \epsilon.
I know...
I was hoping someone could help me understanding winding numbers
For e.g. the point -i that is (0,-1) on this curve...
I was trying to determine if the winding number was 2 or 3
http://img15.imageshack.us/img15/1668/11111111111111countour.jpg
Homework Statement
The principal valueof the logarithmic function of a complex variable is defined to ave its argument in the range -pi < arg(z) < pi. By writing z = tan(w) in terms of exponentials, show that:
tan-1(z) = (1/2i)ln[(1 + iz)/(1 - iz)]
The Attempt at a Solution
I...
[b]1. If f(z) : D--->D is analytic where D is the open unit disk, and
the first (k-1) derivatives at zero vanish i.e (f(0)=0,f'(0)=0,f''(0)=0...f^k-1(0)=0
[b]2.I would like to show that
abs{f(z)} \leq abs{z^k}
[b]3. I believe one can (an the question is...
Hello,
I'm thinking of starting a course in Complex analysis and I'm curious, could one start the course without a deep understanding of analysis of several variables? I know how to do curve integrals and such, partial derivatives, double integrals and all that. What prerequisites are there...
Homework Statement
On an Argand diagram, plot ln(3+4i)
The Attempt at a Solution
ln(3+4i)
= ln(3e2(pi)n + 4ei[(pi)/2 + 2(pi)n]
= i2(pi)n + ln(3+4ei(pi)/2
= ?
Where do I go next with this?
Thanks!
Andrew
Homework Statement
By considering the real and imaginary parts of the product eithetaeiphi, prove the standard formulae for cos(theta+phi) and sin(theta+phi)
Homework Equations
The standard formula for:
cos(theta+phi) = cos(theta)cos(phi) - sin(theta)sin(phi)
sin(theta+phi) =...
Homework Statement
Prove that if f is a meromorphic function f:\mathbb{C}\rightarrow\mathbb{C} with
|f(z)|^5\leq |z|^6\quad\textrm{for all}\quad z\in\mathbb{C}
Then f(z)=0 for all z\in\mathbb{C}
Homework Equations
Liouville's Theorem
A bounded entire function is constant.
The...
Hi,
I cannot work out how the working shown in the attached pic is well, er worked out!:confused:
Could someone explain the ins and outs of the complex analysis of taking the real or imaginary parts of some formula, for example in the context of the my case.
Hi folks,
I have been looking for some time for a video lecture course which deals specifically with complex analysis and think I have covered most of the sources listed in this sub-forum and some in the physics learning materials areas with no luck (including also MIT, YouTube...
Homework Statement
Find two analytic functions f and g with common essential singularity at z=0, but the product function f(z)g(z) has a pole of order 5 at z=0.
Homework Equations
Not an equation per say, but I'm thinking of the desired functions in terms of their respective Laurent...
Homework Statement
Let f(z) = exp(2πiEz) / (1 + z^2), where E is some real number.
Find the poles, their orders and the residues at each pole.
Homework Equations
The Attempt at a Solution
Hi everyone, here's what I've done so far:
1 + z^2 = (1 + i)(1 - i)
Thus f has poles...
Homework Statement
Notation: C=complex plane, B=ball, abs= absolute value, iff=If and only if
Given z0 in C and r>0, determine the path integral along r=abs(z-z0) of the function 1/(z-zo).
2. The attempt at a solution
It seems to me I'm being asked to find the value of a path...
Homework Statement
Show that the function Log(-z) + i(pi) is a branch of logz analytic in the domain D* consisting of all points in the plane except those on the nonnegative real axis.
Homework Equations
The Attempt at a Solution
I know that log z: = Log |z| + iArgz + i2k(pi)...
Homework Statement
I'm trying to prove that \int_0^{\infty}\frac{t^a}{t^b+1}dt=\frac{\pi}{b}\csc\left[\frac{\pi}{b}(a+1)\right] for -1<Re(a)<Re(b)-1.Homework Equations
The Attempt at a Solution
I integrated z^a/(z^b+1) along a positively oriented keyhole contour C. As I took the outer...
I am having a hard time understanding the difference between poles and zeros, and simple poles versus removable poles. For instance, consider f(z)=\frac{z^2}{sin(z)} . we can expand sine into a power series and pull out a z, so doesn't that remove the singularity at z=0? Also, I don't see why...
Please help me with them problems:
1) if z^3=1, show that (1-z)(1-z^2)(1-z^4)(1-z^5)=9, zEC
2) if cos(x)+cos(y)+cos(t)=0, sin(x)+sin(y)+sin(t)=0 show that cos(3x)+cos(3y)+cos(3t)=3cos(x+y+t)
3)show that, the roots the equations (1+z)^(2n) +(1-z)^(2n)=0, nEN, zEC are given by the relation...
Homework Statement
An open set in the complex plane is, by definition, one which contains a disc of positive radius about each of its points. Prove that:
(a) the intersection of two open sets is an open set
(b) the union of arbitrarily many open sets is an open set
Homework Equations...
Homework Statement
Find the radius of convergence of the series
\infty
\sum z/n
n=1
Homework Equations
lim 1/n = 0
n->∞
Radius of convergence = R
A power series converges when |z| < R
and diverges when |z| > R
The Attempt at a Solution
Hi everyone...
Homework Statement
Compute the integral
\oint_{|z|=30}\frac{dz}{z^9+30z+1}
Homework Equations
Residue theorem for a regular closed curve C
\onit_C f(z)dz=2\pi i\sum_k\textrm{Res}(f,z_k)
z_k a singularity of f inside C
The Attempt at a Solution
I'd rather not compute the...
Homework Statement
If f(z) is analytic at a point Zo show that the Conjugate(f(z conjugate)) is also analytic there. (The bar is over the z and the entire thing as well.)
The Attempt at a Solution
I know if a function is analytic at Zo if it is differentiable in some neighborhood...
Homework Statement
Find all entire functions f such that
|f(z)|\leq e^{\textrm{Re}(z)}\quad\forall z\in\mathbb{C}
Homework Equations
\textrm{Re}(u+iv)=u
The Attempt at a Solution
I tried using Nachbin's theorem for functions of exponential type. I also tried using the Cauchy...
Homework Statement
Graph the following in the complex plane
{zϵC: (6+i)z + (6-i)zbar + 5 = 0}
Homework Equations
z=x+iy
zbar=x-iy
The Attempt at a Solution
Substituting the equations gives
2(6x-y) + 5 = 0
=> y = 6x + (5/2)
But that's a line in R^2. The imaginary parts...
Homework Statement
We're supposed to find a bijective mapping from the open unit disk \{z : |z| < 1\} to the sector \{z: z = re^{i \theta}, r > 0, -\pi/4 < \theta < \pi/4 \}.Homework Equations
The Attempt at a Solution
This is confusing me. I tried to find a function that would map [0,1), which...