Complex analysis Definition and 756 Threads

Homework Statement Show that if a function f(z) = u(x,y) +iv(x,y) is entire, then the function conj(f(conj(z))) is entire. Homework Equations (i) The Cauchy-Riemann (CR) equations hold for functions that are entire: u_x = v_y and u_y = -v_x (ii) conj(_) is the conjugate (i.e. there...

Homework Statement Just to make certain that I am understanding this correctly, given a function f(z) = u(x,y) + iv(x,y), the existence of the satisfaction of the Cauchy-Riemann equations alone does not guarantee differentiability, but if those partial derivatives are continuous and the...

Homework Statement Suppose v is a harmonic conjugate of u in a domain D, and that u is a harmonic conjugate of v in D. Show how it follows that u(x,y) and v(x,y) are constant throughout D. The Attempt at a Solution since u is a harmonic conjugate of v, u_xx + u_yy = 0 also, since v...

Homework Statement write f(z)= (z+i)/(z^2+1) in the form w=u(x,y)+iv(x,y) Homework Equations The Attempt at a Solution I tried using the conjugate and also expanding out algebraically but I can not seem to get the right answer. I know what the answer is...

Homework Statement (a) Use the polar form of the Cauchy-Riemann equations to show that: g(z) = ln(r) + i(theta); r > 0 and 0 < (theta) < 2pi is analytic in the given region and find its derivative. (b) then show that the composite function G(z) = g(z^2 + 1) is analytic in the...

In complex analysis, what exactly is the purpuse of the luarent series, i mean, i know that it apporximates the function like a taylor series, an if the function is analytic in the whole domain it simplifies into a taylor series. But i fail to see its purpose - what does it do that the taylor...

Homework Statement Find the radius of convergence of the power series: a) \sum z^{n!} n=0 to infinity b) \sum (n+2^{n})z^{n} n=0 to infinity Homework Equations Radius = 1/(limsup n=>infinity |cn|^1/n) The Attempt at a Solution a) Is cn in this case just 1? And plugging it in...

Homework Statement a) Does f(z)=1/z have an antiderivative over C/(0,0)? b) Does f(z)=(1/z)^n have an antiderivative over C/(0,0), n integer and not equal to 1. Homework Equations The Attempt at a Solution a) No. Integrating over C= the unit circle gives us 2*pi*i. So for at least one...

Homework Statement Show that if c is any nth root of unity other than unity itself that: 1 + c + c^2 + ... + c^(n-1) = 0 Homework Equations 1 + z + z^2 + ... + z^n = (1 - z^(n+1)) / (1 - z) The Attempt at a Solution c is an nth root of unity other than unity itself => (1-c) =/= 0. so, 1 + c...

Homework Statement Homework Equations The Attempt at a Solution How do I go about Q1 and showing the coefficients are unique and then Q2?

I need to calculate the residue of ( 1 - cos wt ) / w^2 This has a pole of second order at w=0, am I correct? Now may math book says that a second order residue is given by limit z goes to z_0 of {[(z-z_0)^2. f(z)]'} where z_0 is the pole I'm quite new to complex...

Homework Statement For each w \in \mathbb{C} define the function \phi_w on the open set \mathbb{C}\backslash \{\bar{w}^{-1}\} by \phi_w (z) = \frac{w - z}{1 - \bar{w}z}, for z \in \mathbb{C}\backslash \{\bar{w}^{-1}\} \back. Prove that \phi_w : \bar{D} \mapsto \bar{D} is a...

Homework Statement Use the residue theorem to compute \int_0^{2\pi} sin^{2n}\theta\ d\theta Homework Equations \mathrm{Res}(f,c) = \frac{1}{(n-1)!} \lim_{z \to c} \frac{d^{n-1}}{dz^{n-1}}\left( (z-c)^{n}f(z) \right) The Attempt at a Solution I started with the substitution z = e^{i\theta}...

Okay, so, I don't understand this concept of 'maximum principle'. A few weeks ago we did Liouville's theorem, which states that any bounded complex function is continuous. Okay... (I can't really imagine the picture of a function which is bounded to be constant, e.g. sin(z) is bounded, at...

Homework Statement Let f:C-> C be an entire bijection with a never zero derivative, then f(z)=az+b for a,b\in CHomework Equations The Attempt at a Solution I'm not sure where to begin with this problem. The only ways I see to attack this are based on somehow showing that f' is bounded and then...

Homework Statement Calculate the integral \int\limits_0^{\infty} \cos{x^2} dx This is the exercise from complex analysis chapter, so I guess I should change it into a complex integral somehow and than integrate. I just don't know how, since neither substitution cos(x^2) = Re{e^ix^2} nor...

I don't think this is the right area to post this question so to the mods: please be kind and move it to a better section if one exists. I'm looking for a textbook on complex analysis which gives proofs but is accessible without a formal real analysis course. I would appreciate suggestions...

Hi everyone! I still didn't fully understand how to get the residues of a complex function. For example the function f(z)=\frac{1}{(z^{2}-1)^{2}} in the region 0<|z-1|<2 has a pole of order 2. So the residue of f(z) in 1 should be given by the limit: \lim_{z \to 1}(z-1)^{2}f(z)=1/4 But...

Homework Statement when complex integral is independent of path? i heard that its for every function f(z) but when i have function f(z)=\left(x^2+y\right)+i\left(xy\right) its not independent, why?

Homework Statement integral: \int\limits_C\cos\frac{z}{2}\mbox{d}z where C is any curve from 0 to \pi+2i The Attempt at a Solution can i do this like in real analysis when counting work between two points, just count this integral and put given data in?

Homework Statement integral: \int\limits_C\frac{\mbox{d}z}{z} where C is circle of radius 2 centered at 0 oriented counterclockwise Homework Equations The Attempt at a Solution I am going to parameter this: \gamma=2\cos t+2i\sin t,\ \gamma^\prime=-2\sin t+2i\cos t,\ t\in[0,2\pi], then...

Suppose that f is entire,= and that f(0)=f'(0)=f''(0)=1 (a) Write the first three terms of the Maclaurin series for f(z) (b) Suppose also that |f''(z)| is bounded. Find a formula for f(z). I believe (a) is just 1+z+(z^2)/2! however (b) I do not know where to begin.

Assume throughout that f is analytic, with a zero of order 42 at z=0. (a)What kind of zero does f' have at z=0? Why? (b)What kind of singularity does 1/f have at z=0? Why? (c)What kind of singularity does f'/f have at z=0? Why? for (a) I'm pretty sure it is a zero of order 41...

Let f(z) be an entire function such that |f(z)| less that or equal to R whenever R>0 and |z|=R. (a)Show that f''(0)=0=f'''(0)=f''''(0)=... (b)Show that f(0)=0. (c) Give two example of such a function f.

Let f be analytic for |z| less than or equal to 1 and suppose that |f(z)| less than or equal to |e^z| when |z|=1. Show (a)|f(z)| less than or equal to |e^z| when |z|<1 and (b)If f(0)=i, then f(z)=ie^z for all z with |z| less than or equal to 1

Homework Statement With f(z) = 2z^{4} +2z^{3} +z^{2} +8z +1 Show that f has exactly one zero in the open first quadrant.Homework Equations Argument PrincipleThe Attempt at a Solution I know I'm supposed to use the Argument Principle.. So far, all I can do is show something like, in the unit...

Hi, I'm a junior undergrad majoring in math and physics, and am deciding between complex analysis and topology for next semester. (I'm planning on doing theoretical physics for grad, something on the more mathematical side, so topology would likely be used). Complex Analysis Pros...

How do you write e^(1/z) in the other form? z = x+yi So we should be able to right it using this definition of e^z, no? e^z = e^x * [cos(y) + i * sin(y)] I pushed some numbers around the page for a while but I can't get 1/(x+i*y) to split into anything nice. Is there a trick?

I just wanted to know what kind of math is needed to solve questions like 1, 2 and 3 of http://www.math.toronto.edu/deljunco/354/ps4.fall10.pdf and number 5 of http://www.math.toronto.edu/deljunco/354/354final08.pdf . I don't need solutions, I just need to know what book or online source can...

How can I express the general wave formula, y=Acos(wt-kx), by the complex ft and the exponential ft? Is it right to use Euler's Identity?

Can anyone suggest a good book on Complex Analysis? I need a book that would be good for self studying.

Suppose f is analytic inside |z|=1. Prove that if |f(z)| is less than or equal to M for |z|=1, then |f(0)| is less than or equal M and |f'(0)| is less than or equal to M. I'm really stuck here on how to approach this problem. Help PLZ!

Suppose that u(x,y) is harmonic for all (x,y). Show that u_x-iu_y is analytic for all z. (Assume that all derivatives in the question exist and are continuous) I have no idea where to start with this? Something with the Cauchy Riemann equations is required but I'm not sure exactly how to...

Homework Statement This is complex analysis by the way. Here's the problem statement:http://i.imgur.com/wegWj.png" I'm doing part b, but some information from part a is carried over. The Attempt at a Solution My problem is that I don't know if I'm being asked to show it via direct...

Hope this does not sound vague! 1) I a looking at the Poisson's formula for the disk. Can somebody give me an example how one uses this, or a question where we use it to solve the problem. What is it exactly saying that Cauchy's formula is not saying? Thank you 2) Can somebody give me an...

Homework Statement For u(x,y)=e^{-y}(x\sin(x)+y\cos(x)) find a harmonic conjugate v(x,y) and express the analytic function f=u +iv as a function of z alone (where z=x+iy0 Homework Equations The Cauchy Riemann equations u_x=v_y and u_y=-v_x and possibly: sin(x) =...

Could someone recommend an accessible and well-known textbook of complex analysis for graduate education? thx

Im a rising junior in the US starting my upper division physics classes. I have an opening this quarter and want to take an applied math course, but cannot decide between these two: In the mathematics department: "Applied complex anlysis Introduction to complex functions and their applications...

What is a good introductory book for complex analysis for self study?

So I will be a sophomore this next semester, and I am having difficulty deciding whether or not to take complex analysis. I am majoring in chemical and biomolecular engineering (with a concentration in cellular/molecular engineering), but I feel after this past semester my heart really lies...

Homework Statement Suppose that f is an entire function. Define g(z)=f*(z*), where * indicates conjugates. I know from another problem that g(z) is also entire. Suppose also that f(z) maps the real axis into the real axis, so that f(x+0i)is in R for at x in R. Show that f(z)=g(z) for all z in...

Homework Statement Q. (a) State Liouville's Theorem (b) Suppose that f is analytic in C and satisfies f(z + m + in) = f(z) for all integers m,n . Prove f is constant. Homework Equations The Attempt at a Solution (a) Liouville's Theorem - If f is bounded and analytic in C, then...

As I am studying for an exam I am trying to wrap my head around the concepts I learned. I want to make sure I fully understand the concepts before the exam in 1.5 weeks. Cauchy's Theorem If u and v satisfy the Cauchy-Riemann equations inside and on the simple closed contour C, then the...

Homework Statement f(z) is a complex function (not necessarily analytic) on a domain D in C. The directional derivative is Dwf(z0)=lim(t->0) (f(z0+tw)-f(z0))/t, where w is a unit directional vector in C. There are three parts to the question: a. Give an example of a function that is not...

Homework Statement The problem is from Sarason, page 44, Exercise IV.14.1. Let f be a univalent holomorphic function in the open connected set G, and let g be the inverse function. Assume that f(G) is open, that g is continuous, and that f\prime\neq 0\forall z\in G. Prove g is...

Homework Statement The problem, for reference, is from Sarason's book "Complex Function Theory, 2nd edition" and is on page 81, Exercise VII.5.1. Let C be a counterclockwise oriented circle, and let f be a holomorphic function defined in an open set containing C and its interior. What is...

Homework Statement This question is in my exam review problem from my complex analysis class. Compute f(100)(0)/100!, where f(z) = 1/(1+i-sqrt(2)z). (f(100)(0) means the 100th derivative of f evaluated at 0.) Homework Equations Cauchy's integral formula might be helpful. The answer to this...

Being a high school student who will be going into physics, should I take complex analysis or abstract algebra in the fall? I can't take both at once, and I am set to take intro to QM (I will already have taken Calc I-III, an introductory functional analysis course, and linear algebra. I also...

Homework Statement This seems to be just a simple limit problem and I feel like I should know it but I'm just not seeing it. I have a continuous function f, and a fixed w I want to show that the limit (as h goes to 0) of the absolute value of: (1/h)*integral[ f(z)-f(w) ]dz = 0...

I'm going to be taking the graduate complex analysis this coming Fall and I've not taken the undergraduate version of the course. It will be a challenge but something that my advisers told me will be surely doable. Anyway, aside from the textbook used for the course, can anyone recommend a...