Complex analysis Definition and 784 Threads

Homework Statement The functions Re(z)/|z|, z/|z|, Re(z^2)/|z|^2, and zRe(z)/|z| are all defined for z!=0 (z is not equal to 0) Which of them can be defined at the point z=0 in such a way that the extended functions are continuous at z=0? It gives the answer to be: Only f(z)=zRe(z)/|z|...

Homework Statement Find the limit points of the set of all points z such that: a.) z=1+(-1)^{n}\frac{n}{n+1} (n=1, 2, ...) b.) z=\frac{1}{m}+\frac{i}{n} (m, n=+/-1, +/-2, ...) c.) z=\frac{p}{m}+i\frac{q}{n} (m, n, p, q=+/1, +/-2 ...) d.) |z|<1 Homework Equations None. The Attempt at a...

Hey guys. I need to calculate this integral so I was thinking about using the residue theorem. The thing is that the point 0 is not enclosed within the curve that I'm about to build, it's on it. Can I still use the theorem? Thanks a lot.

Homework Statement Prove that (z̄ )^k =(z̄ ^k) for every integer k (provided z≠0 when k is negative) Homework Equations The Attempt at a Solution I let z=a+bi so, z̄ =a-bi Then I plugged that into one side of the equation to get (a-bi)^k I was going to try to manipulate this...

*This is not homework, though a class was the origin of my curiosity. In real analysis we could find the equation of a line that passes through two points by finding the slope and then plugging in one set of points to calculate the value of b. ie y = mx + b m = \frac{y_2-y_1}{x_2-x_1} In...

Homework Statement #16)What are the loci of points z which satisfy the following relations...? d.) 0 < Re(iz) < 1 ? g.) α < arg(z) < β, γ < Re(z) < δ, where -π/2 < αα, β < π/2, γ > 0 ? I'm also wondering for help with this proof: #15)...Given: z_1 + z_2 + z_3 = 0 and |z_1| +...

Does anyone have a copy of Saff & Snider's "Fundamentals of Complex Analysis"? So I'm finally, as a graduate student, getting that last piece of undergraduate mathematics I missed: complex analysis. I enrolled in a class at the last minute, and wouldn't you know they assign homework for...

Text: Fundamentals of Complex Analysis With Applications to Engineering and Science by E.B. Saff and A.D. Snider I only ordered my textbook last week (yeah... I know), so I don't think it will get to me before my homework is due. Would some kind soul with this book please post these questions...

Homework Statement Hey guys. Look at this question please. I have two paths, and I need to proof the thing in red. They give a tip due, they say to show first that the equation in green is correct and then using Taylor development, to proof the red equation. I'm still in the first phase...

1. This is something from complex analysis: Find the analytic function f(z)= f(x+iy) such that arg f(z)= xy. 2. w=f(z)=f(x+iy)=u(x,y)+iv(x,y) (*), w=\rho e^{i\theta} (**) Here are the Cauchy-Riemann conditions... \frac{\partial u}{\partial x}=\frac{\partial v}{\partial...

I was just wondering if I was ready to take a 4th year undergrad course in Complex Analysis. The book we will be using is called Complex Function Theory and its buy Sarason. I've taken a course in multivariable calculus, number theory, discrete mathematics, differential equations and modern...

Homework Statement Let Δf= d^2f/dx^2+ d^g/dy^2 (laplace equation - Partial Derivatives) Show Δ(f(g(z))= Mod(g'(z))^2 * Δf(w,v) where g(z)=w(x,y)+v(x,y)i Homework Equations we propably need to use cauchy riemman equations: dw/dx = dv/dy and dw/dy = - dv/dx and chain rule The Attempt...

I'm currently looking for a textbook on Real and Complex Analysis. I currently own both Rudin's and Shilov's, and I'm interested to know if there are any more with that scope of topics. In English, please.

Homework Statement The problem is to integrate: \oint_{C}\frac{dz}{z^{2}-1} C is a C.C.W circle |z| = 2. Homework Equations The Attempt at a Solution I used the Cauchy integral formula: \oint_{C}\frac{f(z)}{(z-z_{0})^{n+1}}dz = \frac{2 \pi i}{n!}f^{n}(z_{0}) Which...

Homework Statement Sketch the locus of |z-2i|=z+3 in C 2. The attempt at a solution Let z=x+iy, then |z-i|=|x+iy-2i)|=|x+i(y-2)|=(x^2+(y-2)^2)^(1/2)=z+3 The problem is that I can't tell what this means geometrically. Is it a spiral?

Can someone please tell me if I have the correct answer for this one? e^(5pi/4) = (1-i)/(-sqrt(2)) Thanks...

I think this should probably be easy, but I am stuck. My book is of no help. Find and describe the locus of points z satisfying the given equations: 1. |z-i|=Re z 2. |z-1|^2 =|z+1|^2 +6 I am thinking for the 1st one that I have to square both sides, but then what? What happens to...

I was reading in a book, says \mu is a measure with compact support K in C, meaning \mu(U)=0 for U\cap K=0.. Is \mu(K) assumed to be finite in this case? It doesn't say in the book, but they make a statement which is true if that's so. Is there usually some assumption about measures being...

hi I want to find an analytic funktion if Re(z) = 1 - x - 2xy My initial thought was to set U(x,y) = 1 - x - 2xy and then solve for V(x,y) through du/dx = dv/dy but it doesn't seem to go as far as I am concernd. Then I thought about the fact that Re(z) = (z + zbar)/2 and then work...

Complex analysis : an introduction to the theory of analytic functions of one complex variable / [by] Lars V. Ahlfors. How do people find it?

Homework Statement Find I = \int_0^{2\pi} \frac{1}{cos\phi+b} d\phi Homework Equations Given above.. The Attempt at a Solution This problem is an introductory problem to trigonometric functions and here is how the answer is obtained - but I have a question about it. First, here...

If f and g are two entire functions such that mod(f(z)) <= mod(g(z)) for all z in C, prove that f=cg for some complex constant c.

I'm not sure whether to post this in the Mathematics or Physics forums, but I figure this problem is easily reduced to its transformation irrespective of the physics it describes. Consider a semi-infinite sheet of (infinitely thin) conductor charged to a potential V. It is placed at a distance...

Hi everyone, hope this is the right place to put this :) I have just finished "Theory of Functions" Vol. 1 & 2 by Konrad Knopp. I'd like to continue with a book that picks up where the second volume it left off. (Especially would be nice is a more "modern" book) The second volume is about...

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...

Homework Statement Let D\subset\mathbb{C} be the unitdisc and F=\{f:D\rightarrow D\,|\,\forall z\in D\partial_{\bar{z}}f=0\}, calculate L=\sup_{f\in F}|f''(0)|. Show that there is an g\in F with g''(0)=L. I am a bit stuck. But I think that it might be an idea to start with Cauchy estimate. Any...

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 Show that f is 2-pi periodic and analytic on the strip \vert Im(z) \vert < \eta, iff it has a Fourier expansion f(z) = \sum_{n = -\infty}^{\infty} a_{n}z^{n}, and that a_n = \frac{1}{2 \pi i} \int_{0}^{2\pi} e^{-inx}f(x) dx. Also, there's something about the lim sup of...

Homework Statement Let f be a holomorphic function in the open subset G or C. Let the point Z of G be a zero of f of order m. Prove that there is a branch of f^(1/m) in some open disk centered at Z Homework Equations Branch- a continuous function g in G such that, for each x in G, the...

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...