Complex analysis Definition and 756 Threads

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...

also P(z)=0, if it is, how is it related to Z=a+bi??

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 does there exist a power series that converges at z= 2+31 and diverges at z=3-i Im really stuck on this one! any ideas?

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)...

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...

is this relashion true? or false? if it is true how can I proof it? (-i)^(-m) = cos((m*pi)/2)+i*sin((m*pi)/2)

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...

Homework Statement I'm supposed to show that, if f is analytic and |f| is constant on a domain D \subset \mathbb{C}, f is constant. Homework Equations The hint is to write f^* = |f|^2 / f. I might also need to use the fact that if f^* is analytic too, then f is constant. The Attempt...

Homework Statement Find a function g analytic in |z|\leq 2, with g(2/3)=0 and |g(z)|= 1 on |z|=2 Homework Equations Bilinear maps B_{\alpha}(z)=\frac{z-\alpha}{1-\overline{\alpha}z} |B_{\alpha}(z)|=1 on |z|=1 The Attempt at a Solution I tried using the maximum...

Homework Statement 1) consider az - b*conj(z) + c = 0 where a,b,c are complex unknown constans express z in terms of a,b,c Homework Equations The Attempt at a Solutionok so i took the conjugate of the original equation to get a second equation: a*conj(z) - b*z + c = 0 so my two...

Homework Statement suppose that f(z) is an analytic function on all of C, and suppose that, for all z in C, we have |f(z)| <= sqrt{|z|} Homework Equations The Attempt at a Solution I'm unsure of how to start the proof. any help is greatly appreciated.

Homework Statement Suppose z0 = x0 + iy0 2 C, and r > 0. Further, suppose that f(z) is a real valued function that is analytic on the open box B(z0; r) = { x + iy | x0 < x < x0 + r; y0 < y < y0 + r }. Then show that f(z) must, in fact, be constant on the box B(z0; r). The Attempt at...

Homework Statement Given |z|<1 and n a positive integer prove that \left|\frac{1-z^n}{1-z}\right|\le n The Attempt at a Solution I try to find the maximum of the function by differentiation \frac{d}{dz}\frac{1-z^n}{1-z}=\frac{-nz^{n-1}*(1-z)+(1-z^n)}{(1-z)^2}=0\Rightarrow...

Homework Statement Hi all. I have the following integral: I = \int_{2 - i\infty}^{2+i\infty}{f(s) \exp(st)ds}, where f(s) is some function. In order to perform this integral, I will choose to close the vertical line with a semi-circle in some halfplane (in order to use Cauchy's integral...

Hi all We look at f(z)=\sqrt z . Here the point z0=0 is a branch point, but can/is z0=0 also regarded as a zero?

Homework Statement Hi all. We we look at z\rightarrow \infty, does this include both z=x for x \rightarrow \infty AND z=iy for y\rightarrow \infty? So, I guess what I am asking is, when z\rightarrow \infty, am I allowed to go to infinity from both the real and imaginary axis? If yes, then this...

Hey, I'm studying Rudin's Real and Complex Analysis by myself and it would be really nice if I could find a solution manual to all/part of the exercises at the end of the chapters. Does anyone know if such a solution manual exists? Thanks

I seem to be missing a subtlety of the definition of a harmonic function. I'm using Churchill and Brown. As stated in the book, an analytic function in domain D with component functions (i.e. real and imaginary parts) u(x,y) and v(x,y) are harmonic in D. harmonic functions satisfy uxx+uyy=0...

How does one show that z^{1/3} is not unique in the complex plane? [ Similarly for z^(1/2) and ln(Z) ] Thanks, Daniel

Homework Statement Hi all. According to my book, a pole z_0 of a function f(z) is defined as \mathop {\lim }\limits_{z \to z_0 } f(z) = \infty. Now let's look at e.g. f(z) = exp(z). Thus we have a singularity for z = infinity, since the limit in this case is infinity. This is what I don't...

Hi all. I have some questions on complex analysis. They are very fundemental. 1) Singularities of a complex functions are the points, where the functions fails to be analytic. Will a singularity then always be a point, where the numerator of the functions is zero? 2) This question is on...

Homework Statement Hi all. My question has to do with integrating rational functions over the unit circle. My example is taken from here (page 2-3): http://www.maths.mq.edu.au/%7Ewchen/lnicafolder/ica11.pdf We wish to integrate the following \int_0^{2\pi } {\frac{{d\theta }}{{a + \cos...

Homework Statement Let B1 = {z element C : abs(z) < 1}, f be a holomorphic function on B1 with Re f(z) > greater than or equal to 0 and f(0) =1. then show that: abs(f(z)) less than or equal to [(1+abs(z))/(1-abs(z))] Homework Equations Schwarz's Lemma: Suppose that f...