Complex analysis Definition and 784 Threads

I saw this method of calculating: $$I = \int_{0}^{1} \log^2(1-x)\log^2(x) dx$$ http://math.stackexchange.com/questions/959701/evaluate-int1-0-log21-x-log2x-dx Can you take a look at M.N.C.E.'s method? I don't understand a few things. Somehow he makes the relation...

Hello, I am evaluating: $$\int_{0}^{\infty} \frac{\log^2(x)}{x^2 + 1} dx$$ Using the following contour: $R$ is the big radius, $\epsilon$ is small radius (of small circle) Question before: Which $\log$ branch is this? I asked else they said, $$-\pi/2 \le arg(z) \le 3\pi/2$$ But in the...

Consider the integral: $$\int_{0}^{\infty} \frac{\log^2(x)}{x^2 + 1} dx$$ $R$ is the big radius, $\delta$ is the small radius. Actually, let's consider $u$ the small radius. Let $\delta = u$ Ultimately the goal is to let $u \to 0$ We can parametrize, $$z =...

Homework Statement First, let's take a look at the complex line integral. What is the geometry of the complex line integral? If we look at the real line integral GIF: [2]: http://en.wikipedia.org/wiki/File:Line_integral_of_scalar_field.gif The real line integral is a path, but then you...

I am looking for a conformal map from a "polygon" to eg the upper half plane, which consists of circle segments instead of lines. So for example, it could be a quadrilateral ABCD, but where AB is a circle segment. The closest I can find is the Schwarz-Christoffel mapping. Anyone has any tips?

Homework Statement Show that the real and imaginary parts of the following susceptibility function satisfy the K-K relationships. Use the residue theorem. $$ \chi(\omega) = \frac{\omega_{p}^2}{(\omega_0^2-\omega^2)+i\gamma\omega} $$ Homework Equations The Kramers-Kronig relations are $$...

I have one slot to fill in in the coming term. The two candidates are Functional Analysis and Complex Analysis (both on the undergraduate level). Here are some questions: 1) Which one would you pick and why? 2) What other classes in the standard B.Sc. math curriculum rely on either of these...

Having just finished an introductory course on group theory (with some bits of ring and field theory), I am completely enthralled with this type of math. I initially planned on taking Complex Analysis next semester since so many people say it's "useful" for physics (this was also a compromise...

Homework Statement r = 2\cos(\theta) Homework EquationsThe Attempt at a Solution Hello, please do not evaluate. Why do textbook state that the derivative of the polar function (symbolic) is dy/dx and not dr/d\theta? It is a function of theta, then why is the derivative dy/dx? Idea: Even...

In complex analysis differentiability for a function ##f## at a point ##z_0## in the interior of the domain of ##f## is defined as the existence of the limit $$ \lim_{h\rightarrow{}0}\frac{f(z_0+h)-f(z_0)}{h}.$$ But why are the possible ##z_0##'s in the closure of the domain of the original...

Homework Statement \int_{-\infty}^{\infty} \frac{\sin(x)}{x} using Complex Analysis Homework Equations Contour analysis on \int_{-\infty}^{\infty} \frac{\sin(x)}{x} The Attempt at a Solution Hello, I am completely new to contour integration. I would really appreciate it if someone can walk...

I'm asked to evaluate the following integral: \int_{c} \frac{30z^2-23z+5}{(2z-1)^2(3z-1)}dz where c is the unit circle. This function has a simple pole at z=\frac{1}{3} and a second order pole at z=\frac{1}{2}, both of which are within my region of integration. I then went about computing the...

Homework Statement Determine the location and type of singularity of f(z) = 1/sin^2(z) Homework EquationsThe Attempt at a Solution I'm not really sure how to calculate this. At this point, we don't have explicit formulae for the coefficients of a Laurent series so I really don't know what to...

Is Complex Analysis and Complex Variables the same thing? Is Complex Analysis pure or applied math? Is Complex Variables pure or applied math? What's the prerequisite of Complex Analysis and Complex Variables? Are they useful for the field of computer science?

Hi there, I'm currently taking Complex Analysis but do not feel like I have enough practice problems or course material (books, websites, Youtube channels, and etc) to study from. I was hoping some of you would have some stuff that I can check out. It would be greatly appreciated. I'm currently...

Hi there, I was just wondering if anyone knows of any good materials, books, websites, Youtube users etc for me to teach myself Complex Analysis for school. Some good practice problems with answers and explanations would be wicked too. Thanks :)

Homework Statement I_1 = \int_0^{2\pi} \frac{sin\theta}{3+2cos\theta} d\theta Homework Equations Using identities to change from cos, sin, to variables of z, I get: 2iz^2 + 6iz + 2i in my denominator The Attempt at a Solution Looking for a singularity, will I use a quadratic...

Hi All, I was reading through Kreyzeig's Advanced Engineering Mathematics and came across two theorems in Complex Analysis. Theorem 1: Let f(z) = u(x,y) + iv(x,y) be defined and continuous in some neighborhood of a point z = x+iy and differentiable at z itself. Then, at that point, the...

Homework Statement Hi, I need to calculate the following integral: \int_{-\infty}^{+\infty}dx \frac{(\pi+\sqrt{x^2+m^2})^2(1+\cos x)}{(x^2-\pi^2)^2\sqrt{x^2+m^2}} The Attempt at a Solution I tried complexifying it: \oint dz...

I came across an interesting problem that I have made no progress on. Let f be an analytic function on the disc ##D = \{z \in C ~|~ |z| < 1\}## satisfying ##f(0) = 1##. Is the following statement true or false? If ##f(a) = f^\prime(a) ## whenever ##\frac{1+a}{a}## and ##\frac{1-a}{a}## are...

I am and EE and CS double major and I am not sure whether to take complex analysis or not. Linear algebra, multivariable calculus, differential equations and probability are compulsory but complex analysis and stochastic processes are optional, so I am wondering whether I should take them or...

I've taken basic undergraduate Real and Complex Analysis, and I've noticed they focus on different kinds of functions. Real analysis studies things like Dirichlet and Cantor functions with infinitely many discontinuities while complex analysis studies mostly differentiable functions. My...

i. Let f and g be functions with a pole at c. Create rules (and prove them) about how we can combine f and g at c. and ii: Find the poles of the function : \frac{cotz+cosz}{sin2z} and classify these poles using part i.

Hi! I am having a hard time with residues. :( I understand the formal definition of a residue, but anything past that and I am struggling. My course lecturer has a very confusing way of organising the course material and it is very hard to comprehend so was wondering if anyone could help with...

Homework Statement Let f(z) = sqrt(z) be the branch of the square root function with sqrt(z) = (r^1/2) (e^iΘ/2), 0≤Θ<2\pi, r > 0 (a) for what values of z is sqrt(z^2) = z? (b) Which part of the complex plane stretches, and which part shrinks under this transformation? Homework...

Homework Statement I can't seem to get a few questions involving inverse trigonometric functions and hyperbolic functions. Here is one that I am stuck on: Evaluate the following in the form x+iy: sinh-1(i/2) = z Homework Equations sinh z = (ez - e-z)/2 The Attempt at a...

Hello, I'm solving the problems given in previous exams, and there's this question: Homework Statement a/ Give the value of ln(i), ln(-i) and i^i b/ If zo=-1-i , what is the value of lim [ ln(zo+e)-ln(zo+i*e) ] when e-> 0 Same question with zo=1+i Homework Equations The...

Homework Statement f(z) = u(x, y) + iv(x, y) where z ≡ x + iy. Let the fluid velocity be V = ∇u. If f(z) is analytic, show that df/dz = V_x − iV_y Homework Equations V_x = du/dx V_y = idu/dy The CR equations du/dx = dv/dy, du/dy = -dv/dx. The Attempt at a Solution I...

Hello, Homework Statement Develop in Fourier series 1/cos(z) and cotan(z) for Im(z)>0 Homework Equations The Attempt at a Solution I really don't know how to do this, i was looking at my notes and we just saw Fourier transform and there is no example for complex functions. I...

Homework Statement After successfully solving a lot of integrals I gathered 4 ugly ones that I can not solve: a) ## \int _{-\infty} ^\infty \frac{cos(2x)}{x^4+1}dx## b) ##\int _0 ^\infty \frac{dx}{1+x^3}## c) ##\int _0 ^\infty \frac{x^2+1}{x^4+1}dx## d) ##\int _0 ^{2\pi } \frac{d\varphi...

Homework Statement Calculate real integrals using complex analysis a) ##\int_{-\infty}^{\infty}\frac{dx}{x^2+1}## b) ##\int_0^\infty \frac{sin(x)}{x}dx##Homework Equations The Attempt at a Solution a) ##\int_{-\infty }^{\infty }\frac{dz}{z^2+1}=\int_{-R}^{R}\frac{dx}{x^2+1}+\int...

How to find the sum using complex analysis sin^3x+sin^32x+sin^33x+sin^34x+...+sin^3nx

I'm about to start scheduling my courses for next year, and I have the option of taking either Real Analysis or Complex Analysis. I'm double majoring in Math and Physics, and I want to go to grad school to study either Applied Mathematics or Physics. I haven't taken any higher level math...

Hello, I'm sorry if I'm not posting this to the correct place - this is my first post on PhysicsForums.com My question regards derivatives of analytic functions. Here it goes: Let w(z) = u(x,y) +iv(x,y) be an analytic function, where z = x + iy, for some x,y that are real...

Homework Statement Show that the inequality\left|\frac{z^2-2z+4}{3x+10}\right|\leq3holds for all z\in\mathbb{C} such that |z|=2 Homework Equations Triangle inequality The Attempt at a Solution I'm not really sure how to go about this. the x is throwing me off. Should I write it out with...

Homework Statement Let a continuous function ##f:\mathbb{C}\rightarrow\mathbb{C}## satisfy ##|f(\mathbb{C})|\rightarrow\infty## as ##|z|\rightarrow\infty## and let ##f(\mathbb{C})## be an open set. Then ##f(\mathbb{C})=\mathbb{C}##. The Attempt at a Solution Suppose for contradiction that...

Hi guys. It's almost time to choose my courses for this year. I'm torn between taking PDE's due to how important it is for physics, or complex analysis due to just liking pure maths. If I do well enough, I'm *possibly* looking to do further study in mathematical physics. I was thinking that if...

I am confused as to what we are obtaining when taking these contour integrals. I know that the close loop contour integral of a holomorphic function is 0. Is this analogous to the closed loop of integral of a conservative force which also gives 0? Also when I am integrating around a...

Homework Statement Compute the first four terms of the Taylor series of \frac{1}{1+e^{z}} at z_{0} = 0 and give it's radius of convergence. Homework Equations e^{z} = \sum\frac{z^{n}}{n!} = 1 + z +\frac{z^{2}}{2!} + \frac{z^{3}}{3!} + o(z^{3}) \frac{1}{1+w} =...

So I am self studying complex analysis using some notes online so here he is trying to figure the map of certain domain applied to a function I follow everything but the end I don't get in the end when he got Re(w) < 1 how did he get that ? is it just because Re(w) which is 1/2 and 0 in this...

[solved]Easy complex analysis question Hi. In the complex plain, since y = 0 (in z=x+iy) at the x axis, shouldn't the following be true? : ##y=0## \int_{-\infty}^{\infty} \frac{\cos(ax)}{x^2+2x+5} dx = \int_{-\infty}^{\infty} \frac{e^{iaz}}{z^2+2z+5} dz = \int_{-\infty}^{\infty} f(z) dz...

Determine the quantitiy of zeroes of the function: f(z)=z^{4}-8z+10 a) Inside the circle | z | < 1 b) Inside the ring 1 \leq | z | < 2a) f(z)=(z^{4}-8z)+10=g(z)+h(z) As |h(z)| \geq |g(z)| \forall z : | z | = 1 Then by Rouche's Theorem the number of zeros of the function inside the circle is the...

Is there a gentle textbook of complex analysis? Something equivalent to Larson's Calculus (or Stewart's). I have Schaum's Outline of Complex Variables (Spiegel-Lipschutz), and it's not bad.

I know that \displaystyle \int_{-\infty}^{\infty}e^{-x^{2}}dx=\sqrt{\pi} (You calculate the square of the integral, combine both integrals, change variable to polar coordinates and you can finally integrate that with ease). But in this exercise I have the following statament: Let be P_{R} the...

Homework Statement Determine the location of the singularities, including those at infinity. For poles also sate the order. f(z) = \frac{1}{(z+2i)^2}-\frac{z}{z-i}+\frac{z+i}{(z-i)^2} Homework Equations Theorem: If a function ##f(z)## has a zero of nth order at ##z_0##, then the function...

Theorem: If a function f(z) has a zero of nth order at z0, then the function h(z)/f(z) has a pole of order n at z0 (where h(z) is analytic at ##z_0##). Can somebody explain this theorem for me? It isn't proved in my book because it's so "easy", but I don't get it? Is the sketch of the proof...

Hello all, I have the following problem from Complex Analysis that I would like for someone to check my understanding on: Homework Statement The problem is to find the derivative if it exists of f(z) = \frac{e^{i\theta}}{r^2} = r^{-2}\cos \theta + i r^{-2}\sin \theta where I have already...

Hi! I'm new here, been a fan of this site for years, but only now I felt the need of registering. Homework Statement Use the Residue Theorem to solve the integral: ∫[(cos(2t)) / (5-4*cos(t)) ] dt from t=0 to t=2pi2. The attempt at a solution I did a variable change z=eit. With that...

Homework Statement If \phi \in \mathcal{M} (group of all linear fractional transformations or Mobius Transformations has three fixed points, then it must be the identity. (The proof should exploit the fact that \mathcal{M} is a group. The Attempt at a Solution Hi all, So...

Homework Statement Show that: Ʃ(-1)n/(n^2+a^2) (from n=0 to ∞) = pi/[asinh(pi*a)], a\neq in, n\in Z. Homework Equations f(z) = f(0) + Ʃbn(1/(z-an)+1/an) (from n=1 to ∞) , where bn is the residue of f(z) at an. The Attempt at a Solution The main problem is I don't how to pick the...