Complex analysis Definition and 755 Threads

Hello guys, Could someone reccomend me a complex analysis textbook? My class is currently using Stewart but i heard this is not such a good text...

1) Let U be a subset of C s.t U is open and connected and let f bea holomorphic function on U s.t. for every z in U, |f(z)| = 1, ie takes takes all points in U to the boundary of the unit circle. Prove that f is constant. Pf. Suppose f is not constant. Then we can find a w s.t. f'(w) is...

Homework Statement Prove that if a function f:c->c is analytic and lim as z to infinity of f(z)/z = 0 that f is constant. Homework Equations Cauchy Integral Formula for the first derivative (want to show this is 0 ie: constant) f prime (z) = 1/(2ipi)*Integral over alpha (circle radius...

Can anyone give me some advice on how to solve this problem? in the reflection principle if f(x) is pure imaginary then the conjugate of f(z)=-f(z*) where z* is the complex conjugate of z. Any advice on where to start? thanks

Can anyone give me some advice on how to solve this problem? in the reflection principle if f(x) is pure imaginary then the conjugate of f(z)=-f(z*) where z* is the complex conjugate of z. Any advice on where to start? thanks

Hello! I am studing for my Complex Analysis exam and solving the exercises for Residues given by the professor. The problem is that for some exercises I get to a solution different from the one of the professor :bugeye:, and I am not sure that the mistake is in my calculations. I would...

Homework Statement 1. Evaluate the following integrals using residues: a) \int _0 ^{\infty} \frac{x^{1/4}}{1 + x^3}dx b) \int _{-\infty} ^{\infty} \frac{\cos (x)}{1 + x^4}dx c) \int _0 ^{\infty} \frac{dx}{p(x)} where p(x) is a poly. with no zeros on {x > 0} d) \int _{-\infty}...

Homework Statement 1. Suppose that f(z) is holomorphic in C and that |f(z)| < M|z|n for |z| > R, where M, R > 0. Show that f(z) is a polynomial of degree at most n. 2. Let f(z) be a holomorphic function on a disk |z| < r and suppose that f(z)2 is a polynomial. Is f(z) a polynomial? Why...

Some hints/help woudl be greatly appreciated! Let I(r) = integral over gamma of (e^iz)/z where gamma: [0,pi] -> C is defined by gamma(t) = re^it. Show that lim r -> infinity of I(r) = 0.

Homework Statement Is there a Laurent Series for Log(z) in the Annulus 0<|z|<1? Homework Equations Go here for the Theorem. It is theorem 7.8: www.math.fullerton.edu/mathews/c2003/LaurentSeriesMod.html[/URL] (copy and paste the link below if you are having problems. Exclude the "[url]" in...

I need a book that's semi-introductory (advanced undergrad to beginning graduate level, if possible) on complex analysis, particularly one that covers power series well, but should be fairly general. I currently have "elementary real and complex analysis" by Georgi Shilov and while it's not...

Preface:The best way I've been taught how to prove that a force is conservative is to take the curl of the force and show that it is equal to zero. That's pretty quick, but after studying for a complex analysis midterm this idea struck my mind. I'm not a master of complex analysis, so there...

Hello everyone, I am trying to solve this follow problem, but don't quite know how to go about getting Arg(z). z = 6 / (1 + 4i) I got that lzl is sqrt((6/17)^2+(-24/17)^2) but am stuck with finding Arg(z). It told me to recall that -pi < Arg(z) <= pi Can you guys teach me how to go...

So my professor threw in what he called an extra 'hard' question for a practice test. So naturally I have a question about it. It relates to the Maximum Modulus Principle: a) Let p(z) = a_0 + a_1 z + a_2 z^2 + ... and let M = max |p(z)| on |z|=1. Show that |a_i|< M for i = 0,1,2. b)...

for our project in calculus, I am doing a presentation on the basics of complex analysis. Somewhere along there I need to tackle the question: what are the applications of complex analysis? Are there any application problems that I can give that involve basic derivatives/integrals of complex...

Hi. So I was reading through "Elementary Real and Complex Analysis" by Georgi E. Shilov (reading the first chapter on Real Numbers and all that "simple" stuff like the field axioms, a bit of set stuff, etc.). Anyways, so while I was reading, I ran into something I couldn't understand... the...

I have two questions on complex integration, and I do not know how to solve them. Please help if you can. Thanks 1. Evaluate the following principal value integral using an appropriate contour. Integration of (integral goes from 0 to infinity) : (x)^a-1/1-x^2, 0<a<1. 2.Using contour...

Let C be a simple closed curve. Show that the area enclosed by C is given by 1/2i * integral of conjugate of z over the curve C with respect to z. the hint says: use polar coordinates i can prove it for a circle, but i am not sure how to extend it to prove it for any given closed curve

Suppose that f is analytic on a domain D, which contains a simple closed curve lambda and the inside of lambda. If |f| is constant on lambda, then either f is constant or f has a zero inside lambda ... i am supposed to use maximum/modulus principle to prove it ... here is my take: if f...

Verify that the linear fractional transformation T(z) = (z2 - z1) / (z - z1) maps z1 to infinity, z2 to 1 and infinity to zero. ^^^ so for problems like these, do I just plug in z1, z2 and infinity in the eqn given for T(z) and see what value they give? in this case, do i assume 1/ 0 is...

(changes in arg h (z) as z traverses lambda)/(2pi) = # of zeroes of h inside lambda + # of holes of h inside lambda now the doubt i have is what happens when the change i get in h (z) is say 9 pi/2 ... because then i would have a 2.5 on left side of the eqn ... so do i round it up and...

find a one-to-one analytic function that maps the domain {} to upper half plane etc ... for questions like these, do we just have to be blessed with good intuition or there are actually sound mathematical ways to come up with one-to-one analytic functions that satisfy the given requirement...

Hi again. Can somebody help me out with this question? "\int_{C_1(0)} \frac {e^{z^n + z^{n-1}+...+ z + 1}} {e^{z^2}} \,dz Where C_r(p) is a circle with centre p and radius r, traced anticlockwise." I'd be guessing that you have to compare this integral with the Cauchy integral formula...

Suppose you have a Meromorphic function f(z) that has a zero at some point in the complex plane. Suppose you create two parallel contours Y1 and Y2 that are parallel and infinitely close to each other yet still contains the zero (the contours are infinitely close to the zero but don't run...

so .. if f (z) = u + iv is analytic on D, then u and v are harmonic on D... now ... if f (z) never vanishes on the domain ... then show log |f (z)| is harmonic on the domain ... Recall: harmonic means second partial derivative of f with respect to x + second partial derivative of f with...

Consider the function: g(z(t)) = i*f '(c+it)/(f(c+it) - a) Where {-d <= t < d} If we let z = c+it By change of variables don't we get: Line integral of g(z(t)) = i ln[f(c+it) - a] evaluated from t = - d to t = d? note: ln is the natural log. Inquisitively, Edwin...

hi, I'm wondering if someone can help me out with this question: "What are the first two non-zero terms of the Taylor series f(z) = \frac {sin(z)} {1 - z^4} expanded about z = 0. (Don't use any differentiation. Just cross multiply and do mental arithmetic)" I know the formula for...

I think I have misunderstood one of the theorems in complex analysis (k reperesents the order of the derivative) Theorem: Suppose f is analytic on a domain D and, further, at some point z[SIZE="1"]0 subset of D, f (k) (z[SIZE="1"]0) = 0. Then f(z) = 0 for all z subset of D ... Is...

Here's my question: Let f and g be analytic inside and on the smple loop \Gamma. Prove that if f(z)=g(z) for all z on \Gamma, then f(z)=g(z) for all z inside \Gamma. Don't really know where to start on this one. This comes from the section 'Cauchy's Integral Formula'.

I'm glad to see that the physics forum website is back online. Suppose you have a function with double poles somewhere on the complex plane. Are there complex analysis techniques that can be used to split the double pole into two single isolated poles? Some example functions might be...

I'm struggling with this question right now: Let the complex velocity potential \Omega(z) be defined implicitly by z = \Omega + e^{\Omega} Show that this map corresponds to (some kind of fluid flow, shown in a diagram, not important). For background, \Omega = \Phi + i\Psi...

Consider a domain D and f:D-->\mathbb{C} a holomorphic function and C a contractible Jordan path contained in D and z1, z2, two points in the interior of C. Evaluate \int_C \frac{f(z)}{(z-z_1)(z-z_2)}dz What happens as z_1 \rightarrow z_2? I have found that \int_C...

Prove that if wz = 0, then w = 0 or z = 0. w and z are two complex numbers. I said that w = a + bi and z = c + di and set wz = 0. I got down to c(a+b) = d(b-a), but don't know where to go from here. I'm trying to teach myself complex analysis, anyone know any good sources? I took the 3...

Suppose you have a unit circle in the complex plane e^{it}, -\infty \leq t \leq \infty. The contour will wind around forever, so at all points in the contour, there are an infinite amount of possible winding numbers, although they are all multiples of 2pi with a well defined contour boundry...

hi there im confused with this question.. Integrate 1) Sin(1/z) dz and 2) Z sin (1/Z^3) where Z is any complex number., over C which is a circle of radius 1 centred at 0 i tried using the cauchy integral formula and stuff but somehow the answer always comes infinity...is...

"Let a,b be in R with a>0 and f(x)=ax^3+bx. Let k(x)=[f''(x)]/[1+(f'(x))^2]^(3/2). Find the critical points of k(x) and use the first derivative test to classify them." This seems incredibly quantitative and complicated for an analysis assignment. There must be a theorem of some kind I can...

Hi there, I'm taking this math for physicists course and we're doing some stuff with functions of complex variables (laurent series residue etc), and I"m having a bit of trouble. I'm not so happy with the book we use. It's a great reference book if you know what you're doing already but...

Can someone tell me when explicit use of complex analysis arises in physics? What particular courses? Thanks

complex analysis-- Oscillation/vibration class Hello all, I'm taking a wave/vibration/oscillation class, and we're delving into complex notation for these. One of our assigments dealt with a complex function that we didn't get a whole lot of practice out of in math methods. I've gone back...

As you know complex analysis has provided many useful tools for harmonic analysis. However I think its application to Einstein's theory of relativity is relatively limited. So I tried to modify complex analysis in order to apply it to the theory of relativity more easily in the following...

"Visual Complex Analysis" I have gotten myself wound around the axel regarding something in "Visual Complex Analysis" (Dr. Tristan Needham) that should be easy. On p. 18 (paperback edition), towards the bottom, the result for two rotations about different points has got me stumped. I cannot...

hello all well i am going to slowly research my way into complex analysis and I decided to start with cauchys theorem i hope this is the best part to start with, well anyway it says that if f(z) is analytic and \frac{f(z)}{z-z_{o}} has a simple pole at z_{0} with residue f(z_{o}) then...

Hi PPls okay i have studied calculus and i can easily see its application in many things like calculating volume,areas,rates ..etc. but i want to know what is the application of complex analysis...where does it all find its uses and why one study it??

Hi all,, I have a problems on complex Analysis: Show that the equation z^4 + z + 5 = 0 has no solution in the set { z is a subset of C: modulus of z is less than 1} i tried doing it using Triangle inequality although i got it but i am looking for a better solution...Pls help

I would appreciate if someone could explain Conformal Mapping using Complex Analysis using an example. I get the rough idea but have no clue how complex analysis comes into the picture. Thank You!

This one is pretty involved so mad props to whoever can help me figure it out. I've been thinking about this for more than an hour and it's bugging me. Consider the function f(z) = 1/(sin(pi/z)). It has singular points at z=0 and z=1/n (where n is an integer). However, my book says each...

I was just wondering what exactly complex analysis is, and what type of applications it's study can be applied to. By the way, an excellent discussion/class on differential forms is taking place here , if anyone would be interested in starting a similar type forum on complex analysis, that...

need some urgent help with basic complex analysis (no proofs) This forum is probably more appropriate. please forgive me for double posting. Can someone give me examples of the following? (no proofs needed) (C is the complex set) 1. a non-zero complex number z such that Arg(z^2) is NOT...

i know it's supposed to be a simple question. frustrating because it is not coming to me. just want a hint. question is: how do you write 1 + cos(theta) + cos (2*theta) + cos(3*theta)... cos(n*theta) using the fact that (z^(n+1) -1) / (z^(n) -1) = 1 + z + z^(2) +... + z^(n) thanks in...

Cauchy integral question The question calls for finding the integral of dz/((z-i)(z+1)) (C:|z-i|=1) I can't figure out how to do this for (C:|z-i|=1). How does this differ from, say, (C: |z|=2) Regards