What is Complex analysis: Definition and 778 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).
Homework Statement
Let Arg(w) denote that value of the argument between -π and π (inclusive). Show
that:
Arg[(z-1)/(z+1)] = { π/2, if Im(z) > 0 or -π/2 ,if Im(z) < 0.
where z is a point on the unit circle ∣z∣= 1
The Attempt at a Solution
First, i know that Arg(w) = arctan(b/a)...
Homework Statement
Sketch the graph
|Re(z)|>2
Homework Equations
z=x+iy
The Attempt at a Solution
|Re(z)|>2
|Re(x+iy)|>2
|x|>2
|x-0|>2, this is a circle centered at zero with radius 2
4. My question
What I'm having a hard time with is the | | notation.
Is this the absolute value, or...
Homework Statement
For f(z) = 1/(1+z^2)
a) find the taylor series centred at the origin and the radius of convergence.
b)find the laurent series for the annulus centred at the origin with inner radius given by the r.o.c. from part a), and an arbitrarily large outer radius.
Homework...
\mathop\int\limits_{\infty} \log[(z-1)(z+1)]dz=A(z)\biggr|_0^0=4\pi i
The infinity symbol below the integral is a positive-oriented, closed, and differentiable path over the function looping around both branch-points and A(z) is the antiderivative of the integrand. I mean would that hold for...
Homework Statement
Use the Definition Re(z1)=Re(z2), Im(z1)=Im(z2)to solve each equation for z=a+bi.
\frac{z}{1+\bar{z}}=3+4i
Homework Equations
Sec 1.1 #42 from Complex Analysis 2nd ed from Dennis Zill
The Attempt at a Solution
I have solved several similar problems like this one in my...
Homework Statement
Consider the equation
(z-1)^23 = z^23
Show that all solutions lie on the line Re(z)=1/2
How many solutions are there
Homework Equations
The Attempt at a Solution
Really have no idea. I figured polar form might be helpful somehow so I converted it and got...
Background: I'm a computer science major, but who has done a lot of math (real analysis, linear/abstract algebra, combinatorics, probab&stats, numerical analysis, linear programming) and currently doing undergraduate research in computational algebra/geometry.
I'm taking a graduate level...
Homework Statement
Explain geometrically why the locus of z such that
arg [ (z-a)/(z-b) ] = constant
is an arc of a certain circle passing through the fixed points a and b.
i tried to visualize the equation in a cartesian co-system but in doing so, i was not very successful.
I have never studied analysis as i am graduate student in engineering. Can anyone point me the elementary book on real and complex analysis preferably junior, undergraduate level book. I found this 2. Can anyone math graduate student comment or put some advice onto it.
1. Elementary Real and...
Hi!
I have to understand how this integral is evaluated (it is taken from Fetter - Quantum theory of many particle systems)(14.24):
http://dl.dropbox.com/u/158338/fis/fetter.pdf"
in particular, i don't know how the log brach cuts are defined..
as far as I know, log branch cuts can be...
Homework Statement
Let z= x + yi be a complex number.
and f(z) = u + vi a complex function.
As:
u = sinx\astcoshy
v= cosx\astsinhy
And if z has a trajectory shown in the attached image.
What would be the trajectory of the point (u,v) ?
Homework Statement
I have to find a conformal map from \Omega = \{ z \in \mathbb C | -1 < \textrm{Re}(z) < 1 \}
to the unit disk D(0,1)
Homework Equations
an analytical function f is conformal in each point where the derivative is non-vanishing
specifically, we can think of linear...
Homework Statement
"Show that there is no conformal map from D(0,1) to \mathbb C"
and D(0,1) means the (open) unit disk
Homework Equations
Conformal maps preserve angles
The Attempt at a Solution
I don't have a clue. I thought the clou might be that D(0,1) has a boundary, and C...
Homework Statement
If an analytic function vanishes on the boundary of a closed disc in its domain
, show it vanishes on the full disc
Homework Equations
CR equations?
The Attempt at a Solution
Not sure how to start this one.
Hope someone could give me some help with a couple of problems.
First:
Proof of -
A function f:G -->Complex Plane is continuous on G iff for every sequence C(going from 1 to infinity) of complex numbers in G that has a limit in G we have
limit as n --> infinity f(C) = f(limit as n...
The title may be a bit vague, so I'll state what I am curious about.
Since complex field is 'extension' to the real field, and in electrodynamics we use things like Stokes theorem, or Gauss theorem, that are being done on real field (differential manifolds and things like that, right?), can...
The part about Laurent series in my Complex Analysis book is somewhat vague and Wikipedia etc. didn't help me much.
I am hoping someone would tell me the exact mathematical definition of a Laurent series (around a given point?) of a given function, perhaps providing an example. Also, how can...
[b]1. Let z be a complex variable. Describe the set of all z satisfying |z^2-z|<1.[\b]
I have a `brute force' solution, but it's really messy. Without a graphing utility, it would be nearly impossible to graph.
I just computed |z^2-z| in terms of x and y, and solved |z^2-z|=1 in this...
hi!.
I have been looking for good complex analysis text.
But, unfortunately, I haven't found it yet.
Could you recommend some complex analysis textbooks except those books whose authers are silverman, alfors, churchill, conway ??
I've been trying to work through this and see whether you can take an "area" in the complex plane, have x,y vary in some interval, and integrate complex functions over that "area."
The math doesn't seem to work out; plus intuitively, if you're going to sum up a function in a complex variable...
Hi all,
I am currently a 2nd year mathematics and physics student. I am working, for the first time, on my own research and just sort of getting my feet wet. I got in touch with a professor that studies Special Functions and he led me to the Legendre functions and associated Legendre...
given the function
arg\xi(1/2+is)
is this an increasing function of 's' ?? , i mean if its derivative is always bigger than 0
here xi is the Riemann Xi function
http://en.wikipedia.org/wiki/Riemann_Xi_function
could we define the 'inverse' (at least for positive s) of...
Homework Statement
i) Find a suitable formula for log z when z lies in the half-plane K that lies above the x-axis, and
from that show log is holomorphic on K
ii) Find a suitable formula for log z when z lies in the half-plane L that lies below the x-axis, and
from that show log is...
I'm a bit lost on this part of my course (ODE's and complex analysis). We've only done about 2-3 of these (seemingly simple) problems where we're given the equation of a line or circle in the complex plane and are asked to find its image in the U-V plane with some transformation \omega, but I...
Hello,
I am wondering what I should brush up on for a class in Complex analysis and Diff Equations. I am planning to take these in the fall and this will be by far the toughest math I will have had. I took a 4 credit Calc II with a solid A. Currently taking Calc III (through Green, Stokes and...
1. Verify that f(z) = Sqrt(z^2 - 1) maps the upper half plane I am z > 0 onto the upper half plane I am w > 0 slit along the segment from 0 to i. [Hint: use the principal branch]
2. Homework Equations
We studied factional linear transformations with T(z) = (a z + b) / (c z + d) , but I...
Homework Statement
solve integral x^3/(e^x-1) with limits from 0 to infinity
Homework Equations
The Attempt at a Solution
i tried using a rectangular contour,the boundaries of the contour pass through z=0 but the complex equivalent has pole at z=0. by Cauchy theorem the function...
Homework Statement
Let f be analytic throught C, suppose that |f(z)|<=M|z|^n for a real constant M and positive integer n. Show that f is a polynomial function of degree less than n.
Homework Statement
Let f:C\rightarrowC be differentiable, with f(z)\neq0 for all z in C. Suppose limf(z) is exist and nonzero as z tends to z0. Prove that f is constant.
[b]1. find the number of solutions of e^iz - z^2n - a = 0 in the upper half of the complex plane, where n is a natural number and a is a real number such that a>1.
[b]2. Rouche's theorem: If f and g are analytic functions in a domain, and |f|>|g| on the boundary of the domain, then the...
S is a star-shaped open subset of \mathbb{C}, f is a holomorphic function from S to \mathbb{C}, z_0 is an element of S.
I've just come out an exam and wondered whether the following 2 statements are true or false:
1 Let g be a holomorphic function on S \subseteq \mathbb{C}, with the...
http://www2.imperial.ac.uk/~bin06/M2...nation2008.pdf
Solutions are here.
http://www2.imperial.ac.uk/~bin06/M2...insoln2008.pdf
My first question is about 3(ii), the proof of Cauchy's integral formula for the first derivative.
The proof here uses the deformation lemma
(from second...
In a lecture today we evaluated a integral:
\oint_{\Gamma} \dfrac{3z - 2}{z^2 - z} dz
Where,
\Gamma = \{ z \in \mathbb{C} | |z| + |z-1| = 3 \}
Our lecturer evaluated it to be 6πi
I sort of understood how he did it, but he really rushed through his steps.
Homework Statement
sketch the curve in the z-plane and sketch its image under w=z^2
|z-1|=1
Homework Equations
z=|z|e^(iArgz)
argw=2argz
The Attempt at a Solution
At first I simply sketched the solution for a circle centered at (1,0) in the z-plane and then mapped that to...
Homework Statement
Evaluate the integral with respect to x from 0 to infinity when the integrand is x^2/(1+x^6), using complex integration techniques.
Homework Equations
The Attempt at a Solution
I have no idea where to start. Please help!
Hello!
I know that the theory of complex analysis is useful to compute integrals of real valued functions. I am a Physics student and I followed a Complex Analysis course but we did not have time to cover this up.
I am looking for a textbook that takes a practical approach to this subject. I...
Homework Statement
The question asks me to find the integral from 0 to infinity of 1/(x^3 + 1), where I have to use the specific contours that they specify. Now I know that I need to use residues (in fact just one here) and the singular point is (1+sqrt(3)*i)/2. Once I can factor the (x^3...
Homework Statement
|a| < 1 a is arbitrary, then show that |z| \leq 1 iff \frac{z-a}{1-a(bar)z} \leq 1
Homework Equations
possible the triangle inequality
The Attempt at a Solution
\frac{z-a}{1-a(bar)z} is analytic everywhere except at 1/a(bar)
|z - a|2 \leq |1-a(bar)z|2...
As the title says, I was wondering what would be a good book in Complex Analysis at the Undergraduate Level? I have one or two of them but like neither of them.
Homework Statement
Find the Maclaurin series representation of:
f(z) = {sinh(z)/z for z =/= 0 }
{0 for z = 0 }
Note: wherever it says 'sum', I am noting the sum from n=0 to infinity.
The Attempt at a Solution
sinh(z) = sum [z^(2n+1)/(2n+1)!]...
hey there
there is this thing we learn in complex analysis (and almost everywhere) that if a function is analytic in a known region, then the integral on a closed path(say, any loop), will be zero.
so there is another statement we need to deal with hear, which is exactly the opposite, that if...
Homework Statement
If f,g are entire functions and |f(z)| <= |g(z)| for all z, draw some conclusions about the relationship between f and g
Homework Equations
none
The Attempt at a Solution
I just need a push in the right direction.. thanks for any and all help!
Homework Statement
Compare the function f(z) = (pi/sin(pi*z))^2 to the summation of g(z) = 1/(z-n)^2 for n ranging from negative infinity to infinity. Show that their difference is
1) pole-free, i.e. analytic
2) of period 1
3) bounded in the strip 0 < x < 1
Conclude that they are...
Homework Statement
Let f and g be two holomorphic functions in the unit disc D1 = {z : |z| < 1}, continuous in D1, which do not vanish for any value of z in the closure of D1. Assume that |f(z)| = |g(z)| for every z in the boundary of D1 and moreover f(1) = g(1). Prove that f and g are the...
Homework Statement
Let f_n be a sequence of holomorphic functions such that f_n converges to zero uniformly in the disc D1 = {z : |z| < 1}. Prove that f '_n converges to zero uniformly in D = {z : |z| < 1/2}.Homework Equations
Cauchy inequalities (estimates from the Cauchy integral formula)The...
Homework Statement
Let f be a holomorphic function in the unit disc D1 whose real part is constant.
Prove that the imaginary part is also constant.
Homework Equations
Cauchy Riemann equations
The Attempt at a Solution
Hi guys, I'm working through the basics again. I think here we...
Homework Statement
We want to create a map from (x,y) to (u,v) such that the right side (positive x) of the hyperbola x^2 - y^2 = 1 is mapped onto the line v = 0 AND all the points to the left of that hyperbola are mapped to above the line. The mapping should be one-to-one and conformal...