Complex analysis Definition and 755 Threads

I've never had any complex analysis, but I'd like to teach myself. I don't know of any good books though. I learned Real Analysis with Pugh, so I'd like a Complex Analysis book on a similar level (or maybe higher). I.e., I'm looking for a book that develops Complex Numbers and functions...

I have a question on complex analysis. Given a differential equation, \dfrac{d^2 \psi}{dx^2} + k ^2 \psi = 0 we know that the general solution (before imposing any boundary conditions) is, \psi (x) = A cos(kx)+B sin(kx). Now here's something I don't quite understand. The solution...

I'm an undergraduate studying mathematics. I did really well in differential equations and abstract algebra, but struggled with our course "Analysis I." I'm taking complex analysis next spring (here's a description of the course, but I'm sure it's not much different than any other complex...

Hi. I need to show that for all |z|\leq1 : (3-e)|z|\leq|e^{z}-1|\leq(e-1)|z| Now...

I have the following problems (1)Consider the series ∑z^n,|z|<1 z is in C I thik this series is absolutely and uniformly comvergent since the series ∑|z|^n is con vergent for |z|<1,but I have a book saying that it is absolutely convergent,not uniformly...i am confused... (2)for the function...

[SOLVED] complex analysis - maximum modulus &amp; analytic function Hi all, I'm having difficulty figuring out how to do the following two problems in complex analysis. I need help! 1. Consider the infinite strip -\pi< I am z < \pi. Does maximum modulus principle apply to this strip? Why or...

[SOLVED] Complex Analysis PROBLEM Let a function f be entire and injective. Show that f(z)=az+b for some complex numbers a,b where a is not 0. Hint: Apply Casorati-Weierstrass Theorem to f(1/z). THEOREM Casorati-Weierstrass Theorem: Let f be holomorphic on a disk D=D_r(z_0)\{z_0} and have an...

Also when trying to find the integral of (1/8z^3 -1) around the contour c=1. I found the singularities to be 1/2, 1/2exp(2pi/3), and 1/2exp(4pi/3) What is the next step here. Do I just assume the integral is 6pi(i) after using partial fractions to find the numerators of the 3 fractions...

Homework Statement Prove that there is no holomorphic function f in the open unit disk such that f(1/n)=((-1)^n)/(n^2) for n=2,3,4... Homework Equations The identity theorem: Let f and g be holomorphic functions in the connected open subset of C, G. If f(z)=g(z) for all z in a subset...

I don't really know which forum to post this in but I just have a quick question: Is it sufficient to say that a function is analytic on a domain if it has a derivative and the derivative is continuous?

Let R be domain which contains the closed circle: |z|<=1, Let f be analytic function s.t f(0)=1, |f(z)|>3/2 in |z|=1, show that in |z|<1 f has at least 1 root, and and one fixed point, i.e s.t that f(z0)=z0. now here what I did, let's define g(z)=f(z)-z, and we first need to show that the...

Homework Statement Let p(z) be a polynomial of degree n \geq 1. Show that \left\{z \in \mathbb{C} : \left|p(z)\right| > 1 \right\}[/tex] is connected with connectivity at most n+1. Homework Equations A region (connected, open set) considered as a set in the complex plane has finite...

[SOLVED] Complex Analysis Show that \mbox{Re}\left(\frac{Re^{i\theta}+r}{Re^{i\theta}-r}\right)=\frac{R^2-r^2}{R^2-2Rr\cos\theta+r^2} where R is the radius of a disc. I was able to show this for all real values of r. However, the problem doesn't specify whether r is real or complex. After...

I think this is the first time I've used this forum for myself. :approve: OK, I'm picking out courses for next semester. Right now I'm in the second semester of Complex Analysis (based on Serge Lang's book) which is a grad level course in single variable complex analysis. My school offers a...

I want to show that the integral from -1 to 1 of z^i = (1-i)(1+exp(-pi)/2 where the path of integration is any contour from z=-1 to z=1 that lies above the real axis (except for its endpoints). So, I know that z^i=exp(i log(z)) and the problem states that |z|>0, and arg(z) is between -pi/2...

Homework Statement \int_{|z| = 2} \sqrt{z^2 - 1} Homework Equations \sqrt{z^2 - 1} = e^{\frac{1}{2} log(z+1) + \frac{1}{2} log(z - 1)} The Attempt at a Solution Honestly, my only thoughts are expanding this as some hideous Taylor series and integrating term by term. But I know...

Complex analysis: having partials is the same as being "well defined?" My professor proved this theorem in class and I don't know if I even wrote it down correctly in my notes. I don't have access to the book so I need to know if this makes sense. Here is the theorem: Under these conditions...

Homework Statement Show that \frac{z}{(z-1)(z-2)(z+1)} has an analytic antiderivative in \{z \in \bold{C}:|z|>2\}. Does the same function with z^2 replacing z (EDIT: I mean replacing the z in the numerator, not everywhere) have an analytic antiderivative in that region? Homework...

Homework Statement Evaluate \int_{\gamma} \sqrt {z} dz where \gamma is the upper half of the unit circle. I contend that this problem does not make sense i.e it is ambiguous because they did not tell us specifically what branch of the complex square root function to use. Am I right?Homework...

Please recommend a complex analysis book for "The road to reality" Guys I am a electrical engineer who studied calculus III about 15 years ago. That time I memorized formulas to pass exams and never have much of a understanding of complex analysis. Never touched high math again after...

I am a physics major and I have taken many math courses, but not Complex Variables. I did a little contour integration along time ago, but I never took it as a course. I do, however, have the option to take this semester. Should I take it instead of another physics elective? I know that it is...

Could anybody please give advice for the study of complex analysis, Riemann surfaces & complex mappings. These subjects form the content of chapters 7 & 8 of Roger Penrose's "The Road to Reality". Any advice will do: maybe suggestions on books to supplement the learning, or books to further my...

I am to find all plints z in the complext plane that satisfies |z-1|=|z+i| The work follows: let z=a+bi |a+bi-1|=|a+bi+i| (a-1)^2+b^2=a^2+(b+1)^2 a^2-2a+1+b^2=a^2+b^2+2b+1 -a=b the correct answer should be a perpendicular bisector of segments joining z=1 and z=-i my result looks...

Homework Statement Write z^3 + 5 z^2 = z + 3i as two real equations Homework Equations z=a+bi? The Attempt at a Solution I've been just playing around with this. I expanded, grouped the real and imaginary parts. I'm really just think I'm groping around desperately in the dark. I think...

I am studying signal processing. I took a class last year but don't have a class now (it is very part time) and would like to do some self study. I did alright in my last class but feel that my appreciation of it would have been greater if I had a better background in complex analysis. Could...

Hello!, I was studing the conformal maps in complex analysis, I don't understand this definition: Definition: A map f:A\rightarrow\mathbb{C} is called conformal at z_0 if there exist a \theta\in[0,2\pi] and r>0 such that for any curve \gamma(t) which is differentiable at t=0, for which...

If Pdx+Qdy is locally exact in R`,prove that \intPdx+Qdy=0 for every cycle Y-0 in R`.

I read the reviews that it's one of the classics in this topic, I wonder does someone know why AMS publishing stopped printing the book (im reffering to the three volumes in one book), especially when the book was published in 2005, i know that it's not profitable but for classic book i would...

Homework Statement f(z) = (z+1)/(z-1) What are the images of the x and y axes under f? At what angle do the images intersect? Homework Equations z = x + iy The Attempt at a Solution This is actually a 4 part question and this is the part I don't understand at all really. The...

I have a homework question that reads: Represent the following rational functions as sums of elementary fractions and find the primitive functions ( indefinite integrals ); (a) f(z)=z-2/z^2+1 But my confusion arrises when I read sums of elementary fractions. I think what the question is...

Homework Statement f:Complex Plane ->Complex Plane by f(z) = (e^z - z^e)/(z^3-1) continuous? (Hint: it has more than one discontinuity.) The Attempt at a Solution My attempt at a solution was thus, initially I expanded z^3 and tried to find where it equaled 1. That wasn't...

So my teacher explained what holomorphic functions were today. But it did not make much sense. As I am attempting to do my homework, I realized that I still don't really know what a Holomorphic function is, let alone how to show that one is. The questions looks like this: show that...

Homework Statement 1) \frac{e^{z}-1}{z} Locate the isolated singularity of the function and tell what kind of singularity it is. 2) \frac{1}{1 - cos(z)} z_0 = 0 find the laurant series for the given function about the indicated point. Also, give the residue of the function at the...

Homework Statement Evaluate \oint_C \f(z) \, dz where C is the unit circle at the origin, and f(z) is given by the following: A. e^{z}^{2} (the z2 is suppose to be z squared) B. 1/(z^{2}-4) Homework Equations The Attempt at a Solution I'm completely confused

Homework Statement Determine where the function f has a derivative, as a function of a complex variable: f(x +iy) = 1/(x+i3y) The Attempt at a Solution I know the cauchy-riemann is not satisfied, so does that simply mean the function is not differentiable anywhere?

Homework Statement 1) Where is f(z)=\frac{sin(z)}{z^{3}+1} differentiable? Analytic? 2) Solve the equation Log(z)=i\frac{3\pi}{2} Homework Equations none really... The Attempt at a Solution For #1 I started out trying to expand this with z=x+iy, but it got extremely messy...

Just wondering, when starting on introductory analysis is it logical to do real analysis before complex variables? My guess is complex analysis uses things from real analysis. I'm doing very basic analysis in calc 2, and not sure if its enough to get by complex.

Homework Statement Let f(z) = \frac{1-iz}{1+iz} and let \mathbb{D} = \{z : |z| < 1 \} . Prove that f is a one-to-one function and f(\mathbb{D}) = \{w : Re(w) > 0 \} . 2. The attempt at a solution I've already shown the first part: Assume f(z_1) = f(z_2) for some z_1, z_2 \in...

Hi all, I'm torn between taking complex analysis or differential geometry at the advanced third year level. Which of these would you consider the easiest to self-learn or the least applicable to the study of theoretical physics? I know that differential geometry shows up in general relativity...

Hi! I am signing up to take my first course in complex analysis this upcoming semester at my university. One of the professors with whom I am interested in taking the class is using Complex Analysis 2nd edition by Bak & Newman and the other one is using Complex Variables & Applications 7th...

Homework Statement Find all solutions to: e^{\tan z} =1, z\in \mathbb{C}Homework Equations z = x+yi \log z = ln|z| + iargz +2\pihi, h\in \mathbb{Z}\log e^{z} = x + iy +2\pihi, h\in \mathbb{Z}Log e^{z} = x + iy The Attempt at a Solution I do not really know how to approach this, I tried to...

Homework Statement Solve cosz = 2i , z\in \mathbb{C} The Attempt at a Solution e^{iz}+e^{-iz} = 4i t=e^{-z} t+t^{-1}=4i \Rightarrow t^{2}-4it+1=0 t = (2 \pm \sqrt{5})i log(e^{-z}) = logt z = x + yi;x,y \in \mathbb{R} log(e^{-z}) = log(e^{-y+ix}) = -y +xi +...

Advice: How do I master complex analysis in 5 weeks? ??! Homework Statement Need to be throughly proficient with th first 7 chapters of saff and snider : fundamentals of complex analysis with engineering applications. Homework Equations egads! there's too many! The Attempt at a...

Is my proof correct for lim_(n-> infty) |z_n| = |z| ? Complex Analysis Homework Statement Show that if lim_{n-> infty} z_n = z then lim_{n-> infty} |z_n| = |z| Homework Equations The Attempt at a Solution Is this correct: lim_{n-> infty} |z_n| = |z| iff Assume...

This really is a question on complex analysis but is about Polchinski's introduction to worlsdheet physics, so I am sure people here will answer this easily. I know it is a very basic question. Polchinski considers a field which is analytic and then says that because of this, one may write it...

Homework Statement the question can be ignored - it involves laplace and Z transforms of RLC ckts. Vc(s) = 0.2 ----------------- s^2 + 0.2s + 1 find the partial fraction equivalent such that it is : -j(0.1005) + j (0.1005) --------------...

I'm currently doing a course in complex analysis and we're using fundamentals of complex analysis, by saff and snider. https://www.amazon.com/dp/0139078746/?tag=pfamazon01-20 And Our problem sets are from the questions at the end of the chapters. I'm finding these questions incredibly hard...

I suppose this is the proper place for this question:) I am learning about conformal field theories and have a question about poles of order > 1. If a conformal transformation acts as z \rightarrow f(z), f(z) must be both invertable and well-defined globally. I want to show that...

Hi could please let me know the Dirichlet's theorem(Complex analysis) ,statement atleast... as stated in John B Comway's book if possible ...I don't have the textbook and its urgent that's why...thank You

Oh god, so confused and panicked today:cry: I know this is a very basic question, but, givin the function 1/(z-w)^4 does this have one pole of order 4, or possibly 4 poles of order 1...? Also, could you please clarify, ''to get the zero's of a function, set the numerator = 0'' ''to...