Complex analysis Definition and 756 Threads

Homework Statement Let Omega = C\((-inf,-1]U[1,inf)), find a holomorphic bijection phi:omega-->delta, where delta is the open unit disk Homework Equations Reimann Mapping Theorem Special Mapping formulas: can map wedges onto wedges, with deletion of real line from zero to infinity in...

Hi, I am interested in taking a complex analysis course. How useful is it to the physical sciences?

Homework Statement Evaluate (1/2ipi)* contour integral of [z^(n-1)] / [(3z^n) - 1 ] dz Homework Equations I would assume you would have to use the Argument Theorem since this problem comes after the proof of the argument theorem in my book. The Attempt at a Solution z^(n-1)...

Homework Statement C = positively oriented simple closed piecewise smooth path Prove that: (1/2i)*\int_{C}\bar{z}dz is the area enclosed by C. Homework Equations *I know that the curve C is piecewise smooth so that it can be broken up into finitely many pieces so that each piece...

This is addressed to people who know complex analysis (hope this is the right section). Here's the Laurent theorem from my book for my later reference: Suppose a function f is analytic throughout an annular domain R1<|z-z0|<R2, centered at z0, and let C denote any positively oriented simple...

Homework Statement if an entire function satisfies f(z+i)=f(z) and f(z+1)=f(z), must the function be constant? Homework Equations The Attempt at a Solution It's true that f(0) = f(k) = f(ik) where k is an integer. I'm wondering whether I can apply Liouville's theorem into this...

Homework Statement Prove that there does not exist an analytic function on the annulus D: 1<|z|<2, s.t. F'(z) = 1/z for all z in D. Homework Equations The Attempt at a Solution Assume F exists, then for z in D, not a negative number, F(z) = Log z + c since Log' z = 1/z... Lost

Homework Statement integral 1/(a+cos(t))^2 from 0 to pi. Homework Equations cos(t)=1/2(e^it+e^-it) z=e^it dz/(ie^it)=dt The Attempt at a Solution int dt/(a+cos(t))^2 = int dz/iz(a2+az+az-1+z2/4 +1/2 +z-2/4) so with these types of problems I normally can factor this guy...

Homework Statement Show that if the analytic function w= f(z) maps a domain D onto a portion of a line, then f must be constant throughout D. Homework Equations The Attempt at a Solution I just have one question, can I write w = u(x,y) (a+bi) since it maps to a portion of a line...

What are the implications for holomorphicity of a function being a multifunction. take f(z)=\ln{z}=\ln{r}+i arg(z), here z=z_0+2k \pi all correspond to the same value of z but give different values of f(z) i.e. its a multifunction. how does this affect its holomorphicity? as far as i...

Ok, so I'm suppose to be able to remove the singularity to find the residue of the function (z)cos{\frac{1}{z} I tried to see how "bad" the singularity was by taking the limit, but I can't figure out if \lim_{ z \to 0 } (z)cos{\frac{1}{z} goes to 0 or if it is...

Hey, I am looking for a good book on complex analyis (complex calculus, "complexe anlysis" in german). Any recommendations? I am a first year Electrical engineering student at the ETH Zürich. It should cover the following, and have a reasonalbe amount of examples: Analytical Funktions...

Homework Statement Hey guys. I have this question, I took it from a test. I need to check if there is an analytic function F(z) in this area (in the pic) that has this derivative (in the pic). http://img256.imageshack.us/img256/7826/25453238.jpg Well, the derivative is analytic in...

Homework Statement Let f and g be two holomorphic functions in a connected open set D of the plane which have no zeros in D; if there is a sequence an of points such that lim an = a and an does not equal a for all n, and if f'(an)/f(an)=g'(an)/g(an) show that there is a constant c such that...

two questions here: (i) my notes say that \frac{1}{e^{\frac{1}{z}}-1} has an isolated singularity at z=\frac{1}{2 \pi i n}, n \in \mathbb{Z} \backslash \{0\} i can't see this though... (ii) let b \in \mathbb{R}. show \int_{-\infty}^{\infty} e^{-x^2} \cos{(2bx)} dx = e^{-b^2}...

Let w_1,w_2 \in \mathbb{C} and \gamma be some smooth curve from w_1 to w_2. Find \int_{\gamma} e^{\sin{z}} \cos{z} dz this is holomorphic on the entire copmlex plan so we can't use a residue theorem. furthermore, we can't assume \gamma is a closed contour as we aren't told w_1=w_2 so it...

I'm a physics major and I have space for one more class the coming fall semester: either advanced mathematics for engineers and scientists or applied complex analysis. Advanced Mathematics for Engineers and Scientists- Vector analysis, Fourier analysis and partial differential equations...

I am trying to understand theorem 1.17 in page 15-16 international edition 1987. How do you show that \phi_n(t) is a monotonic increasing sequence of functions?

Hi, I just finished up a Complex Analysis course last term and, though I'm no physics major, I thought Quantum Physics looked interesting. Does anyone know some common or interesting applications of Complex Analysis within Quantum Physics? Or even an online resource that might delve into...

Homework Statement Hey guys. I have this integral, I tried to use trigo, tried to use the complex expression but nothing worked, can I please have some help? Thanks a lot. Homework Equations The Attempt at a Solution

Homework Statement Evaluate the following integral for 0<r<1 by writing \cos\theta = \frac{1}{2}(e^{i\theta} + e^{-i\theta}) reducing the given integral to a complex integral over the unit circle. Evaluate: \displaystyle{\frac{1}{2\pi}\int_0^{2\pi}\frac{1}{1-2r\cos\theta +...

Homework Statement Hey guys. So, I need to calculate this integral, I uploaded what I tried to do in the pic. But according to them, this is not the right answer, according to them, the right answer is the one I marked in red at the bottom. Any idea where this Sin came from? Thanks...

Homework Statement Hey guys. So, I need to calculate this integral. I uploaded what I tried to do. First of all, did the substitute, then I tried to use the residue theorem so I was looking for the residue of z=0 which is happen to be a removable singular point so it's just 0, then I...

Homework Statement Hey guys. I have this problem, I need to show that it's true and I don't have a clue. I tried to do like alpha = x+yi but it got me nowhere, any ideas? Thanks. Homework Equations The Attempt at a Solution

Hi, I'm studying complex analysis right now, I would like to use this thread to ask questions when I read books. Many questions will be very stupid, so please bear with me. Also, English is my second language. text: Complex Analysis (2nd edition) author: Stephen D. Fisher [question deleted]...

Hi everyone, I'm a biochemistry major hoping to go into BME (ideally PhD). Besides taking a bunch of extra math courses, I made a list of engineering and intermediate level physics classes that grad schools seem to be looking for, and also kind of figured out what courses offered by my school...

Homework Statement Suppose P is a polynomial such that P(z) is real iff. z is real. Prove that P is linear. The hint given in the text is to set P = u + iv, z = x+iy and note that v = 0 iff y = 0. We are then told to conclude that a. either v-sub y(partial of v with respect to y) is...

Homework Statement Show that the vector z1 is parallel to z2 if and only if Im(z1z2*)=0 note: z2* is the complement of z2 Homework Equations The Attempt at a Solution I would probably convert z to polar form. so, z1=r1(cos Ѳ1+isin Ѳ1) z2=r2(cos Ѳ2+isin Ѳ2) so...

Homework Statement I want to show z_{1}+(z_{2}+z_{3})=(z_{1}+z_{2})+z_{3} with the use of a graph. Homework Equations The Attempt at a Solution I am just cluless on how to graph. I know z=x+iy where the real part is on the x-axis and the imaginary part is on the y axis.

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...

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...