Complex analysis Definition and 784 Threads

Homework Statement If f,g are entire functions and |f(z)| <= |g(z)| for all z, draw some conclusions about the relationship between f and g Homework Equations none The Attempt at a Solution I just need a push in the right direction.. thanks for any and all help!

Homework Statement Compare the function f(z) = (pi/sin(pi*z))^2 to the summation of g(z) = 1/(z-n)^2 for n ranging from negative infinity to infinity. Show that their difference is 1) pole-free, i.e. analytic 2) of period 1 3) bounded in the strip 0 < x < 1 Conclude that they are...

Homework Statement Let f and g be two holomorphic functions in the unit disc D1 = {z : |z| < 1}, continuous in D1, which do not vanish for any value of z in the closure of D1. Assume that |f(z)| = |g(z)| for every z in the boundary of D1 and moreover f(1) = g(1). Prove that f and g are the...

Homework Statement Let f_n be a sequence of holomorphic functions such that f_n converges to zero uniformly in the disc D1 = {z : |z| < 1}. Prove that f '_n converges to zero uniformly in D = {z : |z| < 1/2}.Homework Equations Cauchy inequalities (estimates from the Cauchy integral formula)The...

Homework Statement Let f be a holomorphic function in the unit disc D1 whose real part is constant. Prove that the imaginary part is also constant. Homework Equations Cauchy Riemann equations The Attempt at a Solution Hi guys, I'm working through the basics again. I think here we...

Homework Statement We want to create a map from (x,y) to (u,v) such that the right side (positive x) of the hyperbola x^2 - y^2 = 1 is mapped onto the line v = 0 AND all the points to the left of that hyperbola are mapped to above the line. The mapping should be one-to-one and conformal...

Homework Statement Let f(z) = u(x,y) + iv(x,y) be a continuous, non-constant function that is analytic on some closed bounded region R. Prove that the component function u(x,y) reaches a minimum value on the boundary of R. The Attempt at a Solution By the minimum modulus principle...

Homework Statement Suppose f(z) is entire and the harmonic function u(x,y) = Re[f(z)] has an upper bound u_0. (i.e. u(x,y) <= u_0 for all real numbers x and y). Show that u(x,y) must be constant throughout the plane. The Attempt at a Solution Since f(z) = u(x,y) + iv(x,y) is entire...

What undergraduate math classes would you want to take if you wanted to be exposed to the stuff used in complex analysis? (Besides complex analysis)

I have to integrate S cos^8 (t) dt from 0 to 2 pi, presumably using complex analysis I got to S [(e^(it) + e(-it))/2]^8 dt from 0 ti 2pi How do I take it from here? I have a hint- use binomial theorem.

Homework Statement I'm not very good with LaTeX and the reference button seems to broken. So Assume lim h(z) = 1+i, as z->w, prove there exists a delta, d>0 s.t. 0<|z-w|<d -> (2^.5)/2 < |h(z)| < 3(2^.5)/2 Homework Equations The Attempt at a Solution Kinda been running in...

Its question 1(g) in the picture. My work is shown there as well. This has to do with independence of path of a contour. Reason I am suspicious is that first there is a different answer online and second it says "principal branch" which I have not understood. Does that mean a straight line for...

I was hoping someone could clarify the idea of a branch cut for me. In class, my professor talked about how a branch cut is used to remove discontinuities. He gave an example of |z|=1 needing a branch cut along the positive real axis. If this because going from 0 to 2\pi, the 0 and 2\pi match up?

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...

Homework Statement Find the radius of convergence of \sumcnz^{n} if c2k = 2^{k} and c2k-1 = (1+1/k)^{k^{2}}, k = 1, 2, 3... Homework Equations 1/R = limsup as n=> infinity |cn|^1/nThe Attempt at a Solution I'm not really sure where to start with this. I know that it's a power series, and to...

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 Show that if the polynomial p(z)=anzn+an-1zn-1+...+a0 is written in factored form as p(z)=an(z-z1)d1(z-z2)d2...(z-zr)dr, then (a) n=d1+d2+...+dr (b) an-1=-an(d1z1+d2z2+...+drzr) (c) a0=an(-1)rz1d1z2d2...zrdr Homework Equations Taylor form of a polynomial? p(z) = \sum...

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...

Homework Statement Show that Abel's Limit Theorem holds as the complex number z approaches 1 if instead of taking the requirement |1 - z| \leq M|1-|z|| , you restrict z to |arg(1- z)| \leq \alpha where 0 < \alpha < \frac{\pi}{2} Homework Equations The Attempt at a Solution The latter...