Complex variables Definition and 112 Threads

The lectures haven't been difficult, but the book is hard for me to read. I usually understand math by working problems, but this course doesn't seem to work like that. I just don't know how to complete the exercises in the book. The book is "An Introduction to Complex Analysis and Geometry"...

Homework Statement Suppose that c is a member of the Real numbers, and p is a member of the Complex numbers with p not equal to 0, are given numbers. (a) Show that pz + conjugate(pz) + c = 0 is the equation of a straight line in the plane. Provide a carefully-drawn plot that...

i'm sorry if this is the wrong area to be posting such a question, but I am registered for a complex analysis course starting in the fall and I was wondering what i should review before starting this course? I've had math coursework up through calculus, differential equations, linear algebra...

Homework Statement \int_{0}^{\infty }\frac{dx}{1+x^{5}} The attempt at a solution This is for my complex variables class, so I have been trying to compute it using residues. I noticed that if we extend it to the complex plane and integrate over the edges of a half disk or a quarter...

Homework Statement Find the angle through which a curve drawn from the point z0 is rotated under the mapping w=f(z), and find the corresponding scale factor of the transformation. z0 = -1, w=z^2 and z0 = -1 + i, w = 1/z Homework Equations I honestly don't know how to begin...

Homework Statement Use the definition of a limit to show that lim (z^{2} +c) = z_{0}^{2} +c as {z->z_{0}}Homework Equations Definition of a limit: |f(z)-L|< epsilon if 0<|z-z0|< delta The Attempt at a Solution |(z^{2}+c )-(z_{0}^{2}+c)| = | z^{2}-z_{0}^{2}|= |(z-z_{0})(z+z_{0}) | <...

Find a (complex) polynomial function f of x and y that is differentiable at the origin, with df/dz = 1 at the point z=0, and differentiable at all points on the unit circle x^2 + y^2=1, but is not differentiable at any other point in the complex plane. (Bruce Palka, Page 101) I think we use...

Homework Statement How do I change sin^2(z) to x+iy form? (z=x+iy) I have to put this x and y to arctan(y/x) Homework Equations The Attempt at a Solution I tried to use sin^2(z) = 1/2 -1/2(cos(2z)) or sin(z) = ((e^(iz) - e^(-iz))/2i)^2 but both ways I cannot take out i. Or isn't the...

Homework Statement "This is an example from my textbook: Solve the equation z4 - 4z2 + 4 - 2i = 0 Solution: Rearranging, we get z4 - 4z2 + 4 = 2i or (z2 - 2)2 = 2i = (1+i)2 This has solutions z2 - 2 = 1+i or -1-i. Equivalently z2=3+i or z2=1-i These may be solved to give the 4...

Homework Statement z is a complex number different from 1 and n >= 1 is an integer 1 + z + z^2+ ... + z^n = \frac{z^{n+1} - 1}{z-1} show that: \sin(\theta) + \sin(2 \theta)+ ... \sin(n \theta) = \frac{ \sin(n \theta/2) \sin((n+1) \theta / 2)}{\sin(\theta / 2)} The Attempt at a Solution...

Can anyone recommend a good textbook for an undergrad who has done real and complex analysis and wants to learn about several complex variables? Thanks!

If z is complex, the following rules are true, right? \frac{d}{dz}z^n = nz^{n-1} \frac{d}{dz}\ln{g(z)} = \frac{1}{g(z)} \frac{d}{dz}g(z) \frac{d}{dz}e^{g(z)}=e^{g(z)}\frac{d}{dz}g(z) These are of course the same rules as for real variables. When do I need to be careful about...

Homework Statement map the function \begin{equation}w = \Big(\frac{z-1}{z+1}\Big)^{2} \end{equation} on some domain which contains z=e^{i\theta}. \theta between 0 and \pi Hint: Map the semicircular arc bounding the top of the disc by putting $z=e^{i\theta}$ in the above formula. The...

Hello, I have come across this problem in my studies where I need to try to come up with a graph of a function involving two complex numbers. I have been trying to figure this out for a while now, but I am not sure how to do it. Is there any way to do this type of thing by hand or in Maple...

Homework Statement {Q 6.2.2 from Arfken "Mathematical Methods for Physicists"} Having shown that the real part u(x,y) and imaginary part v(x,y) of an analytic function w(z) each satisfy Laplace's equation, show that u(x,y) and v(x,y) cannot have either a maximum or a minimum in the interior of...

Homework Statement evaluate: \int_{-\infty}^{\infty}e^{-ax^2}cos(bx)dx Homework Equations The Attempt at a Solution since \cos(bx) is the real part of: e^(b*x*i), we can rewrite the integral as: Real[\int_{-\infty}^{\infty}e^{bxi-ax^2}dx] but now I am stuck because I...

Homework Statement I'd like some help with 2 problems: Show by using Demoivre's theorem and the geometric series formula that the sum of all n values of z^(1/n) is zero when n >=2. Z is a complex number. Use the geometric series formula and Demoivre's theorem to show that: Homework Equations...

Hi, I need two simple proofs of complex inequalities. 1) |1-z|/|z|<2 2)|1+z|/|z|>1/2 Ik need them for a bound of a complex integral.It's not homework Thank you

I just completed a course on complex variables. I really enjoyed the application sections. I was thinking of studying CV a little more on my own. The question is: is it worth it to study more CV for physics and engineering? What advantage would it give me? Aside: I was browsing through...

Homework Statement Let z be a complex variable Suppose f is an entire function and Re(f(z))\leq c for all z Show that f is constant. (Hint: Consider exp(f(z)) Homework Equations possibly this: e^z=e^x(cos(y)+isin(y)) where z=x+iy The Attempt at a Solution I had no idea how I...

Homework Statement let: f(z)=u(x,y)+iv(x,y) I want to express the following function like the one above: f(z)=cos(z)\equiv=\frac{1}{2}(e^{iz}+e^{-iz}) Homework Equations (i=sqrt(-1)) f(z) is a complex function The Attempt at a Solution...

Given f(z) = (z+1) / (z-1) for z not equal to 1 My teacher wrote |f(z)| = |x+1+iy| / |x-1+iy| = sqrt((x+1)^2 +1) / sqrt((x-1)^2 + 1) How do the values within the modulus work out to the right hand side? I can't figure it out.

For z not equal to 1 f(z) = (z + 1) / (z - 1) How do you show the function maps {z ϵ C : Re(z) < 0} into {w ϵ C : |w| < 1} and {w ϵ C : |w| < 1} into {z ϵ C : Re(z) < 0}? ---- I don't even know how to start this one besides that "into" means 1-1. How do you show the mappings?

Homework Statement Determine the nature of the singularity at z=0Homework Equations f(z)=\frac{1}{cos(z)}+\frac{1}{z} The Attempt at a Solution by expanding into series: f(z)=\Sigma_{n=0}^{\infty} \frac{(2n)! (-1)^n}{x^{2n}} + \Sigma_{n=0}^{\infty} (-1)^n (z-1)^n Now \frac{1}{z} has no...

Homework Statement Sketch the set of points determined \mid 2\bar z + i \mid = 4. Homework Equations \mid z - z_0 \mid = r and \mid \bar z \mid = \mid z \mid The Attempt at a Solution I know that it will be the circle with radius = 4 and z_0 is the center of the circle and that...

Homework Statement The nth order Chebyshev polynomial is defined by Tn(x)= cos( n arccos(x) ) , n is a positive integer; -1<= x <= 1. Using the de Moivre theorem, show that Tn(x) has the polynomial representation Tn(x)= 1/2 [(x+sqrt(x2-1))n+(x-sqrt(x2-1))n] The Attempt at a Solution I really...

Could someone recommend some good textbooks of complex variables function. I am a physics student with poor mathematical knowledges, so I want to study some mathematical subjects again, by myself. thanks!

Homework Statement Suppose f(z) is an analytic function on domain D, and suppose that, for all z in D, we have 2*Re(f(z)) + 3*Im(f(z))=12. prove that f(z) must be a constant. Homework Equations The Attempt at a Solution ok, I am drawing somewhat of blank with this one but I am...

Homework Statement i) Define a path \gamma whose image is the ellipse \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 traced counterclockwise. ii) Show that \int \frac{1}{z} dz = \int \frac{1}{z} dz for a suitable circle \beta (NOTE: THE FIRST INTEGRAL IS OVER THE ELLIPSE \gamma, THE SECOND ONE IS...

I'm taking a complex variables course, and I'm really stuck at it, I've never felt this way in any math course before :S, I'm starting to get angry. Anyway here is the problem, I hope someone can give me a hand. I believe this is a basic and simple problem in the subject... Homework Statement...

Which course is more difficulty in terms of which subject contains more rigorous proofs, Complex variables or Real analysis. I don't know whether I should dropped Complex variables, but the only reason I am taken it is because of the useful physics applications found in this course. I my...

Homework Statement Let S be the open set consisting of all points such that |z|<1 or |z-2|<1 . State why S is not connected. Homework Equations The Attempt at a Solution According to my complex variables book the definition of a connected set are pairs of points that can be...

Homework Statement Using the fact that |z(1)-z(2)| is the distance between two points z(1) and z(2) , give a geometric argument that a)|a-4*i| + |z+4*i| =10 represents an ellipse whose foci are (0,4) and(0,-4). Homework Equations Triangle inequality equation; distance formula...

Does their book have exercises? tanks Laura

Homework Statement Integral from 0 to 2*pi of [(Cos(3*theta)) / (5 - 4Cos(theta))] (d*theta) Homework Equations z=exp(i*theta), Cos(theta) = (z + z^(-1)) / 2, Cos(3*theta) = ((exp(3*i*theta)+exp(-3*i*theta)) / 2 = (z^(3) + z^(-3)) / 2, dz = (i*z) (d*theta) The Attempt...

Homework Statement Let C. denote the circle |z-z.|= R, taken counter clockwise. use the parametric representation z= z. + Re^(io) (-pi </= o </= pi) for C. to derive the following integration formulas: integral C. (dz/(z-z.)) = 2ipi Homework Equations note: z. and C. represent z knot...

Can anyone help me with this pls? How can you prove that the integral of f(z) around the contour z= 1 is 0 where f(z) is Log(z+5) Thx I know Log(z) is ln r + i (theta). But i don't know how that applies to this situation. Also, do I solve it as a normal integral or use...

Homework Equations Let w \in C be a fixed complex number with |w|<1. Let f(z)=\frac{z-w}{1-\bar{w}z}. Calculate its maximum value in the region |z| \leq 1. The Attempt at a Solution How should I approach this? Not sure if maximum modulus principle is of much help.

Homework Equations Using complex analysis, compute \int_{-\infty}^{\infty} \frac{e^{itx}}{1+x^2}dx where t is real. The Attempt at a Solution I'm not good at complex analysis at all and am totally lost. I do know some Fourier analysis though and using it I got \pi e^{-|t|}. How should I...

I'm thinking of taking an introduction to complex variables course as an elective for my applied math major(or minor). Are their any applications of complex variables in physics?

Homework Statement Let f be analytic in the disk |z| <= 1. Prove that for any 0 < r < 1, |f(0)|^2 <= \frac{1}{\pi r^2} \int \int_{x^2 + y^2 <= r^2} |f(z)|^2 dxdy Homework Equations The hint is apply the Gauss mean value theorem on f^2(z)The Attempt at a Solution Having difficulty starting...

[SOLVED] Complex Variables. Problem about complex sine. Homework Statement Proof that the function \begin{displaymath} \begin{array}{cccc} f: &A=\left\{z\in\mathbb{C}\mid-\frac{\pi}{2}<\Re z<\frac{\pi}{2}\right\} &\longrightarrow &B=\mathbb{C}-\left\{z\in\mathbb{C}\mid...

Homework Statement Let \Omega be a bounded domain in C whose boundary is a curve z = z(t), a<=t<=b, and let A(\Omega) be the area of \Omega. Prove that A(\Omega) = \frac{1}{2}\int^b_a |z(t)|^2 Im(\frac{z'(t)}{z(t)})dt Homework Equations The Attempt at a Solution Not even sure...

Homework Statement Let {z1,z2,z3} be complex variables such that |z1| = |z2| = |z3|. Prove that z1,z2,z3 are vertices of an equilateral triangle iff z1 + z2 + z3 = 0. Homework Equations The Attempt at a Solution Not really sure where to start on this. I know that |z2-z1= |z3-z2|...

If I have a function f(z) such that limit z --> infinity is finite what do I know about that function? Is it "bounded" or am I still confused about what "bounded" means?

Does \int 1/x dx = ln|x| + c in complex variable theory? Or can we relax the absolute value restraint of ln|x|? (as in, can it be ln(x) + c?)

I think I just flunked out on the in-class portion of my complex variables midterm. I have a take home exam to work on over the weekend, and it's not going to be easy. I studied for days for this midterm, the prof said it would be on "proofs" so I learned every proof in the book. But, I wish I'd...

I'm studying the proof of the ML inequlity from complex analysis. I don't know what they did in one step of the proof and I was wondering if anyone can explain the step to me. First of all the theorem says: Let f(z) be continuous on a contour C. Then |\int_{c}f(z)dz| \leq ML Where L...

I can't think of how to title the problem I'm having, but this is what the course is called. Complex being imaginary numbers, ie z = a + ic where i is the sqrt of -1. So here is the question that I have no idea where to start with: Construct a sequence {zn} which is bounded and for...

Hi. I am studying on complex variables on my own using Brown and Churchill, 6th edition. so far i am doing pretty good, but i have run into trouble doing a few contour integrals. 1. First off, the authors state that unlike in calculus, where the integral can mean area under the curve, in...