Complex analysis Definition and 784 Threads

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics; as well as in physics, including the branches of hydrodynamics, thermodynamics, and particularly quantum mechanics. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering.As a differentiable function of a complex variable is equal to its Taylor series (that is, it is analytic), complex analysis is particularly concerned with analytic functions of a complex variable (that is, holomorphic functions).

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  1. PhysicsKid0123

    What Is Conformal Mapping in Complex Analysis?

    "Definition: A map ƒ: A ⊂ ℂ→ ℂ is called conformal at z0, if there exists an angle θ ∈[0,2Pi) and an r > 0 such that for any curve γ(t) that is differentiable at t=0, for which γ(t)∈ A and γ(0)= z0, and that satisfies γ ' ≠0, the curve σ(t) = ƒ(γ(t)) is differentiable at t=0 and, setting u =...
  2. A

    Graduate course as a UG: Complex Analysis or Topology?

    As an undergraduate, which graduate-level course will prepare me better for grad school, Complex Analysis or Topology? I probably can't fit both into my schedule, but I can definitely fit one. I have already taken undergraduate complex analysis and I'm taking now undergraduate topology. My...
  3. L

    Proving that a subspace of C is a field

    Homework Statement Show that the only m for which the subspace of C given by {z ∈ C: Im(z) = m Re(z)} is a field is m=0. Homework Equations Field axioms The Attempt at a Solution I tried to prove one direction : - If z is in the subspace, Re z>0 and m≠0 then Arg z<Arg z^2, so z^2 is not in...
  4. D

    Contour integration & the residue theorem

    When one uses a contour integral to evaluate an integral on the real line, for example \int_{-\infty}^{\infty}\frac{dz}{(1+x)^{3}} is it correct to say that one analytically continues the integrand onto the complex plane and integrate it over a closed contour ##C## (over a semi-circle of radius...
  5. Remixex

    Calculus Beginners Guide to Complex Analysis

    OK, so i took a course named "Oscillations and vibrations" We began the course with an "introduction" to complex numbers, basically we raced through them in like 3 classes, we talked about how to get complex roots, adding, multiplying, Cauchy-Riemann Conditions, Cauchy's integration Theorem...
  6. A

    Why isnt Cauchy's formula used for the perimeter of ellipse?

    So the formula for an ellipse in polar coordinates is r(θ) = p/(1+εcos(θ)). By evaluating L = ∫r(θ) dθ on the complex plane on a circle of circumference ε on the centered at the origin I obtained the equation L = (2π)/√(1-ε^2). Why then does Wikipedia say that the formula for the perimeter is...
  7. PhysicsKid0123

    I need advice on upcoming course schedule

    So I really have a few questions. First, is it wise to take the following classes in the same semester? Quantum Mechanics I - At the level of Griffiths' Intro to Quantum Mechanics, chapters 1-5ish Classical Dynamics - At the level of Thorton and Marion, Chapters 1-12, ends with coupled...
  8. Titan97

    Analysis Books on complex analysis and algebra

    can you recommend a good book on complex analysis? I would like a book that can sharpen my skills in solving complex number problems through graphs and also improve the algebraic part like solving problems related to roots of unity etc. (I have studied calculus myself. I have done a lot of self...
  9. ion santra

    What is the Nature of Singularity in the Function f(x)=exp(-1/z)?

    what is the nature of singularity of the function f(x)=exp(-1/z) where z is a complex number? now i arrive at two different results by progressing in two different ways. 1) if we expand the series f(z)=1-1/z+1/2!(z^2)-... then i can say that z=0 is an essential singularity. 2) now again if i...
  10. RJLiberator

    Complex Analysis Contour Circle Question

    Homework Statement I have uploaded necessary image(s) for the question I have successfully accomplished a, but I am not sure how to start b. Homework Equations The sum of the integral paths added up = the desired result. The Attempt at a Solution [/B] So we start with path CR And then go...
  11. RJLiberator

    Complex Analysis Integral Question

    Homework Statement Computer the integral: Integral from 0 to infinity of (d(theta)/(5+4sin(theta)) Homework Equations integral 0 to 2pi (d(theta)/1+asin(theta)) = 2pi/(sqrt(1-a^2)) (-1<a<1) The Attempt at a Solution I've seen this integral be computed from 0 to 2pi, where the answer is 2pi/3...
  12. RJLiberator

    Complex Analysis Clarification Question

    Homework Statement Problem and solution found here: http://homepages.math.uic.edu/~dcabrera/math417/summer2008/section57_59.pdf The question I am interested in is #1. In the solution, the instructor differentiates the series to get to: 2/(1-z)^3 = the series. If I want the Maclaurin series of...
  13. RJLiberator

    Complex Analysis Series Question

    Homework Statement Let 0 < r < 1. Show that from n=1 to n=∞ of Σ(r^ncos(n*theta)) = (rcos(theta)-r^2)/(1-2rcos(theta)+r^2) Hint. This is an example of the statement that sometimes the fastest path to a “real” fact is via complex numbers. Let z = reiθ. Then, since r = |z|, and 0 < r < 1, the...
  14. M

    Roots of Negative Numbers (Complex Analysis)

    Homework Statement Express (-1)1/10 in exponential form (My first time posting - I hope I got the syntax right!) Homework Equations The Attempt at a Solution [/B] I got the solution, it's ejπ/10, but I'm not sure why. Here's my work: (-1)1/10 = (cos(π) + jsin(π))1/10 = cos(pi/10) +...
  15. O

    Can a Non-Constant Holomorphic Function Equal Zero Everywhere?

    Homework Statement With . Give an example, if it exists, of a non constant holomorphic function that is zero everywhere and has the form 1/n, where n € N. Homework Equations So.. This was in my Complex Analysis exam, and i have no idea what to do. I always seem to get stuck at these more...
  16. RJLiberator

    General Understanding of Terms [ Complex Analysis ]

    Hi all, I was unsure where to put this thread as I read the main topic title in the topology/analysis forums and decided to post it here. I am looking for a chart/graph/website that helps me understand the basic terms such as: -neighborhoods -Boundary points -Singularity points - "Function is...
  17. RJLiberator

    Simple Complex Analysis Clarification

    I am currently learning how to work with Cauchy-Riemann equations. The equation is f(z) = 2x+ixy^2. My question: is u(x,y) = 2x or just x? At this link: http://www.math.mun.ca/~mkondra/coan/as3a.pdf in letter e) they say u(x,y) is equal to x. But I don't understand how that is possible. Is...
  18. RJLiberator

    Complex Analysis simple Mapping question

    Homework Statement Find the image of the rectangle with four vertices A=0, B= pi*i, C= -1+pi*i, D = -1 under the function f(z)=e^x 2. The attempt at a solution So, the graph of the original points is obvious. Now I have to map them to the new function. Seems easy enough, but I am not getting...
  19. RJLiberator

    Complex Analysis Properties Question 2

    The problem states, Show that: a) |e^(i*theta)| = 1. Now, the definition of e^(i*theta) makes this |cos(theta)+isin(theta)| If we choose any theta then this should be equal to 1. What can help me prove this? If I choose, say, pi/6 then it simplifies to |(sqrt(3))/2+i/2)| which doesn't seem to...
  20. RJLiberator

    Complex Analysis Properties Question

    Use properties to show that: (question is in the attached picture) Now, it is my understanding that due to properties you can express (sqrt(5)-i) as the sqrt((sqrt(5))^2+(-1)^2) which equals sqrt(6). And (2zbar+5) can be represented as (2z+5). But this would be sqrt(6)*(2z+5) which is NOT...
  21. L

    Where to Find Challenging Complex Analysis Problems Online?

    Sorry if this is the wrong forum to post this- Can anyone suggest a good (ideally online) resource for challenging complex analysis problems? The ones I have found so far have been mainly computational- I'm looking for conceptually harder problems, preferably requiring lots of proofs, which...
  22. M

    Use Residue Theorems or Laurent Series to evaluate integral

    Homework Statement Evaluate the integral using any method: ∫C (z10) / (z - (1/2))(z10 + 2), where C : |z| = 1 Homework Equations ∫C f(z) dz = 2πi*(Σki=1 Resp_i f(z) The Attempt at a Solution Rewrote the function as (1/(z-(1/2)))*(1/(1+(2/z^10))). Not sure if Laurent series expansion is the...
  23. RJLiberator

    Should I take Complex Analysis? Am I ready?

    Here's my situation: Summer 2015, I am majoring in math and physics. I am taking a 4-week course on DIFF EQ right now, and completely loving it and doing extremely well. Just finished my set of Calc 1, 2, and 3, and an intro to advanced math course (proof-writing basics). Diff EQ is a 220...
  24. R

    Derivatives and Linear transformations

    Let G be a non-empty open connected set in Rn, f be a differentiable function from G into R, and A be a linear transformation from Rn to R. If f '(a)=A for all a in G, find f and prove your answer. I thought of f as being the same as the linear transformation, i.e. f(x)=A(x). Is this true?
  25. N

    Complex Analysis: Use of Cauchy

    http://www.math.hawaii.edu/~williamdemeo/Analysis-href.pdf Please look at problem 2 on page 39 of the problems/solutions linked above. I know I'm going to kick myself when someone explains this to me but how was equation "(31)" of the solution obtained? The first term of the RHS of (31) is...
  26. E

    Complex analysis book recommendation for electrical engineering

    I need recommendation about complex analysis book. As I'm electrical eng. student, it should cover everything one engineer need to know about that mathematical field, but without strict mathematical formalism :)
  27. N

    Complex Polynomial of nth degree

    Homework Statement Show that if P(z)=a_0+a_1z+\cdots+a_nz^n is a polynomial of degree n where n\geq1 then there exists some positive number R such that |P(z)|>\frac{|a_n||z|^n}{2} for each value of z such that |z|>R Homework Equations Not sure. The Attempt at a Solution I've tried dividing...
  28. N

    Complex Analysis: Contour Integral

    Here's a link to a professor's notes on a contour integration example. https://math.nyu.edu/faculty/childres/lec12.pdf I don't understand where the ##e^{i\pi /2} I## comes from in the first problem. It seems like it should be ##e^{i\pi}## instead since ##-C_3## and ##C_1## are both on the real...
  29. Coffee_

    Complex analysis and vector fields

    I'm going to ask a very general question where I just would want to hear different possible methods that can be thought of in this kind of problem. I am trying to solve a very specific problem with this but I won't talk about that because I don't want someone to give me the answer but ideas for...
  30. K

    2nd order pole while computing residue in a complex integral

    Hello, I am trying to understand how to get the residue as given by wolfram : http://www.wolframalpha.com/input/?i=residue+of+e^{Sqrt[x^2+%2B+1]}%2F%28x^2+%2B+1%29^2 The issue I am facing is - since it is a second order pole, when I try to different e^{\sqrt{x^+1}} I get a \sqrt{x^+1}...
  31. M

    Integrals of the function f(z) = e^(1/z) (complex analysis)

    How do you integrate f(z) = e^(1/z) in the multiply connected domain {Rez>0}∖{2} It seems like integrals of this function are path independent in this domain since integrals of e^(1/z) exist everywhere in teh domain {Rez>0}∖{2}. Is that correct?
  32. N

    Complex Analysis: Open Mapping Theorem, Argument Principle

    Homework Statement In each case, state whether the assertion is true or false, and justify your answer with a proof or counterexample. (a) Let ##f## be holomorphic on an open connected set ##O\subseteq \mathcal{C}##. Let ##a\in O##. Let ##\{z_k\}## and ##\{\zeta_k\}## be two sequences...
  33. M

    Path dependence (Complex Analysis)

    Are the integrals of the function f(z) = (1/(z-2) + (1/(z+1) + e^(1/z) path independent in the following domain: {Rez>0}∖{2} The domain is not simply connected I know that path independence has 3 equivalent forms that are 1) Integrals are independent if for every 2 points and 2 contours...
  34. M

    Is f(z) = e^(z^2) for all z in C?

    Assume |f(z)| >= 1/3|e^(z^2)| for all z in C and that f(0) = 1 and that f(z) is entire. Prove that f(z) = e^(z^2) for all z in C. How do you start for this.
  35. N

    What Is the Standard Name for This Theorem About Meromorphic Functions?

    Hi, In my textbook the following theorem is designated "Proposition 3.4.2 part (vi)". There are 6 parts total in the overall theorem. I'll just type the part I'm interested in below. My question is, is there a more standard name for this theorem? I would like to find an additional...
  36. N

    Factoring equation with real coefficients

    Homework Statement Find the roots of z^4+4=0 and use that to factor the expression into quadratic factors with real coefficients. Homework Equations DeMoivre's formula. The Attempt at a Solution I have been able to identify they are \pm 1 \pm i but i have no idea how to factor the...
  37. DrPapper

    Exploring Mary Boas' Theorem III: Analytic Functions & Taylor Series

    On page 671 Mary Boas has her Theorem III for that chapter. Roughly it tells us that if f(z) -a complex function- is analytic in a region, inside that region f(z) has derivatives of all orders. We can also expand this function in a taylor series. I get the part about a Taylor series, that's...
  38. N

    Complex Analysis: Series Convergence

    Homework Statement For ##|z-a|<r## let ##f(z)=\sum_{n=0}^{\infty}a_n (z-a)^n##. Let ##g(z)=\sum_{n=0}^{\infty}b_n(z-a)^n##. Assume ##g(z)## is nonzero for ##|z-a|<r##. Then ##b_0## is not zero. Define ##c_0=a_0/b_0## and, inductively for ##n>0##, define $$ c_n=(a_n - \sum_{j=0}^{n-1} c_j...
  39. N

    Complex Analysis: Identity Theorem

    Homework Statement Let f be a function with a power series representation on a disk, say D(0,1). In each case, use the given information to identify the function. Is it unique? (a) f(1/n)=4 for n=1,2,\dots (b) f(i/n)=-\frac{1}{n^2} for n=1,2,\dots A side question: Is corollary 1 from my...
  40. N

    Complex Analysis: Special Power Series

    Homework Statement Give an example of a power series with [itex]R=1[\itex] that converges uniformly for [itex]|z|\le 1[\itex], but such that its derived series converges nowhere for [itex]|z=1|[\itex]. Homework Equations R is the radius of convergence and the derived series is the term by term...
  41. M

    Complex Analysis: Largest set where f(z) is analytic

    Homework Statement Find the largest set D on which f(z) is analytic and find its derivative. (If a branch is not specified, use the principal branch.) f(z) = Log(iz+1) / (z^2+2z+5) Homework EquationsThe Attempt at a Solution Not sure how to even attempt this solutions but I wrote down that...
  42. N

    What is the Theorem for Differentiability in Advanced Calculus?

    Homework Statement This isn't a standard homework problem. We were asked to do research and to find a theorem of the form: If something about the partial derivatives of u and v is true then the implication is ##D(u,v)## at ##(x_0,y_0)## exists from ##R^2## to ##R^2##Homework EquationsThe...
  43. M

    Branch points [Complex Analysis]

    Homework Statement Hi, I'm stuck with this question: How many branches (solutions) and branch points does the function f(z) = (z2 +1 +i)1=4 have? Give an example of a branch of the multi- valued function f that is continuous in the cut-plane, for some choice of branch cut(s). Now by choosing...
  44. M

    Set of Points in complex plane

    Homework Statement Describe the set of points determined by the given condition in the complex plane: |z - 1 + i| = 1 Homework Equations |z| = sqrt(x2 + y2) z = x + iy The Attempt at a Solution Tried to put absolute values on every thing by the Triangle inequality |z| - |1| + |i| = |1|...
  45. A

    Residue of f(z) involving digamma function

    Homework Statement Find the residue of: $$f(z) = \frac{(\psi(-z) + \gamma)}{(z+1)(z+2)^3} \space \text{at} \space z=n$$ Where $n$ is every positive integer because those $n$ are the poles of $f(z)$Homework EquationsThe Attempt at a Solution This is a simple pole, however: $$\lim_{z \to n}...
  46. A

    Find the residue of g(z) at z=-2 using Laurent Expansion

    Homework Statement Find the residue at z=-2 for $$g(z) = \frac{\psi(-z)}{(z+1)(z+2)^3}$$ Homework Equations $$\psi(-z)$$ represents the digamma function, $$\zeta(z)$$ represents the Riemann-Zeta-Function. The Attempt at a Solution I know that: $$\psi(z+1) = -\gamma - \sum_{k=1}^{\infty}...
  47. A

    How to deal with this sum complex analysis?

    Homework Statement Homework Equations Down The Attempt at a Solution As you see in the solution, I am confused as to why the sum of residues is required. My question is the sum: $$(4)\cdot\sum_{n=1}^{\infty} \frac{\coth(\pi n)}{n^3}$$ Question #1: -Why is the beginning n=1 the residue...
  48. neosoul

    Complex numbers and differential equations for physics

    How relevant is complex analysis to physics? I really want to take differential equations but I would have to change my schedule around way more than I want to. So, would anyone advise a physics major to to take complex analysis? Should I just change my schedule around so I can take differential...
  49. A

    Replacing Variables in Integration

    Homework Statement $$I = \int_{-\infty}^{\infty} e^{-x^2} dx$$ Homework Equations Below The Attempt at a Solution $$I = \int_{-\infty}^{\infty} e^{-x^2} dx$$ I don't understand, we say: $$I = \int_{-\infty}^{\infty} e^{-x^2} dx$$ Then we say: $$I = \int_{-\infty}^{\infty} e^{-t^2} dt$$...
  50. A

    Calculating Harmonic Sums using Residues

    I posted the same question on Math Stackexchange: http://math.stackexchange.com/questions/1084724/calculating-harmonic-sums-with-residues/1085248#1085248 The answer there using complex analysis is great. I had questions, which Id like to get advice on here. (1) How did he get the laurent...
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