What is Complex variables: Definition and 120 Discussions

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. M

    Drawing sets of Complex Variables

    I tried saying z = x + iy, then squared both sides so that I would get something that looked like: |z - i|^2 + |z + i|^2 + |z - i||z + i| = 3, where the first two terms are simple but the third term is what I don't know what to do with. I'm wondering if I'm using the wrong approach. For that...
  2. Measle

    Complex Analysis - sqrt(z^2 + 1) function behavior

    Homework Statement Homework Equations The relevant equation is that sqrt(z) = e^(1/2 log z) and the principal branch is from (-pi, pi] The Attempt at a Solution The solution is provided, since this isn't a homework problem (I was told to post it here anyway). I don't understand why the...
  3. Measle

    I Confused by the behavior of sqrt(z^2+1)

    (mentor note: this is a homework problem with a solution that the OP would like to understand better) In Taylor's Complex Variables, Example 1.4.10 Can someone help me understand this? I don't know what they mean by (i, i inf), or how they got it and -it
  4. Measle

    I Principal branch of the log function

    I'm learning complex analysis right now, and I'm reading from Joseph Taylor's Complex Variables. On Theorem 1.4.8, it says "If a log is the branch of the log function determined by an interval I, then log agrees with the ordinary natural log function on the positive real numbers if and only if...
  5. K

    Complex Variables: Continuity

    Homework Statement Determine if the following function is continuous: f(x) = (x-iy)/(x-1) Homework Equations How do find out if a function is continuous without graphing it and without a point to examine? I know I've learned this, probably in pre-calculus too, but I'm blanking The Attempt at...
  6. I

    Courses How difficult is complex variables?

    Hello, I am a rising sophomore in Astronomy and Physics. I am taking complex variables next semester and was wondering the effort required to succeed in the class. There are some other classes I'd like to take, however I don't want to overload myself. I have taken up through multivariable calc...
  7. B

    Calculating Integral Using Residue Theorem & Complex Variables

    Homework Statement I have never formally studied complex analysis, but I am reading this paper: http://adsabs.harvard.edu/abs/1996MNRAS.283..837S wherein section 2.2 they make use of the residue theorem. I am trying to follow along with this (and have looked up contour integration, cauchy's...
  8. J

    MHB Complex Variables - Max Modulus Inequality

    Suppose that f is analytic on the disc $\vert{z}\vert<1$ and satisfies $\vert{f(z)}\vert\le{M}$ if $\vert{z}\vert<1$. If $f(\alpha)=0$ for some $\alpha, \vert{\alpha}\vert<1$. Show that, $$\vert{f(z)}\vert\le{M\vert{\frac{z-\alpha}{1-\overline{\alpha}z}}\vert}$$ What I have: Let...
  9. J

    MHB Complex Variables - Solution of a System

    Suppose the polynomial p has all its zeros in the closed half-plane $Re w\le0$, and any zeros that lie on the imaginary axis are of order one. $$p(z)=det(zI-A),$$ where I is the n x n identity matrix. Show that any solution of the system $$\dot{x}=Ax+b$$ remains bounded as $t\to{\infty}$...
  10. J

    MHB Complex Variables - Legendre Polynomial

    We define the Legendre polynomial $P_n$ by $$P_n (z)=\frac{1}{2^nn!}\frac{d^n}{dz^n}(z^2-1)^n$$ Let $\omega$ be a smooth simple closed curve around z. Show that $$P_n (z)=\frac{1}{2i\pi}\frac{1}{2^n}\int_\omega\frac{(w^2-1)^n}{(w-z)^{n+1}}dw$$ What I have: We know $(w^2-1)^n$ is analytic on...
  11. J

    MHB Complex Variables - Zeros of Analytic Functions

    Studying for my complex analysis final. I think this should be a simple question but wanted some clarification. "Extend the formula $$\frac{1}{2i\pi} \int_\omega \frac{h'(z)}{h(z)}\, dz = \sum_{j=1}^N n_j - \sum_{k=1}^M m_k$$ to prove the following. Let $g$ be analytic on a domain...
  12. K

    Complex Analysis. Laurent Series Expansion in region(22C).

    <Moderator's note: moved from a technical forum, so homework template missing> Hi. I have solved the others but I am really struggling on 22c. I need it to converge for |z|>2. This is the part I am really struggling with. I am trying to get both fractions into a geometric series with...
  13. E

    Courses What Math Course is Best Paired with Linear Algebra?

    I'm currently an applied math major. I'm creating a schedule for my next semester and I have the choice to take either complex variables or vector analysis with linear algebra and a college geometry course(elective of choice), but I don't know which pairing will be less stressful. I am currently...
  14. P

    Contour integral- Complex variables

    Homework Statement evaluate ##\int \frac{sinh(ax)}{sinh(\pi x)}## where the integral runs from 0 to infinity Homework Equations The Attempt at a Solution consider ##\frac{sinh(az)}{sinh(\pi z)}## Poles are at ##z= n \pi i## So I'm considering the contour integral around the closed contour...
  15. naima

    Fourier transform with complex variables

    I found this formula in a paper: \int exp( \frac{x1 + i x2}{ \sqrt 2} \eta^* - \frac{x1 - i x2}{ \sqrt 2} \eta) D(\eta)/ \pi d^2 \eta the author calls it the Fourier transform of D. It is the first time thar i see this formula. How common is this notation? Can we use it without problem?
  16. N

    Proof using hyperbolic trig functions and complex variables

    1. Given, x + yi = tan^-1 ((exp(a + bi)). Prove that tan(2x) = -cos(b) / sinh(a) Homework Equations I have derived. tan(x + yi) = i*tan(x)*tanh(y) / 1 - i*tan(x)*tanh(y) tan(2x) = 2tanx / 1 - tan^2 (x) Exp(a+bi) = exp(a) *(cos(b) + i*sin(b))[/B] 3. My attempt...
  17. S

    Complex variables and classical mechanics

    Dear all, I'd like to know what is the place/use of complex variables (and complex analysis) in classical mechanics. By the way, is there any? Thanks for your help. Best regards!
  18. H

    Is f(z)+i arg(z) an analytic function?

    Is this function analytic or not? Please explain
  19. S

    Doubt in Partial derivative of complex variables

    Today, I had a class on Complex analysis and my professor wrote this on the board : The Laplacian satisfies this equation : where, So, how did he arrive at that equation?
  20. K

    Jacobi elliptic functions with complex variables

    I am trying to solve a Duffing's equation ##\ddot{x}(t)+\alpha x(t)+\beta x^3(t)=0## where ##\alpha## is a complex number with ##Re \alpha<0## and ##\beta>0##. The solution can be written as Jacobi elliptic function ##cn(\omega t,k)##. Then both ##\omega## and ##k## are complex. The solution to...
  21. Y

    Complex Variables - principal argument

    Homework Statement Find the principle argument Arg z when z = (sqrt(3) - i)^6 Homework Equations The Attempt at a Solution I'm sorry to say that I'm not sure how to solve this problem. It's my understanding that what this question is basically asking me to do is find theta such that...
  22. 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|...
  23. S

    Does (-2)^(⅔) have an imaginary component?

    Some calculators say (-2)2/3 is equal to ##-\frac{1}{2^\frac{1}{3}}+i\frac{3^\frac{1}{2}}{2^\frac{1}{3}}## while others say its equal to ##4^{\frac{1}{3}}## i.e. ##|-\frac{1}{2^\frac{1}{3}}+i\frac{3^\frac{1}{2}}{2^\frac{1}{3}}|##. I think I am right to imply from above that (-2)2/3 does have an...
  24. V

    Topology of Relativity: Implications of Niels Bohr's Arguments

    I have seen in the online Stanford Encyclopedia of Philosophy in the entry on Copenhagen Interpretation of Quantum Mechanics that Niels Bohr had argued that the theory of relativity is not a literal representation of the universe: "Neither does the theory of relativity, Bohr argued, provide us...
  25. M

    Are Complex Analysis and Complex Variables the same thing?

    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?
  26. B

    Laurent series expansion of Log(1+1/(z-1))

    Homework Statement Find the Laurent series expansion of f(z) = \log\left(1+\frac{1}{z-1}\right) in powers of \left(z-1\right). Homework Equations The function has a singularity at z = 1, and the nearest other singularity is at z = 0 (where the Log function diverges). So in theory there should...
  27. T

    Power Expansion (Complex variables)

    Homework Statement Use the power series for e^z and the def. of sin(z) to check that sum ((-1)^k z^(2 k+1))/((2 k+1)!) Homework Equations The Attempt at a Solution I apologize, but I am not particularly good with latex. Therefore, I attached a picture of my solution thus far...
  28. R

    Can someone explain this equality to me (complex variables)

    Homework Statement I hate to upload the whole problem, but I am trying to evaluate an indefinite integral, and I can follow the solution until right near the end. The example says that for a point on C_R|e^{-3z}|=e^{-3y}\leq 1. I don't understand how they can say this. Below is the question...
  29. E

    Taylor Series for Complex Variables

    Homework Statement Obtain the Taylor series ez=e Ʃ(z-1)n/n! for 0\leq(n)<\infty, (|z-1|<\infty) for the function f(z)=ez by (ii) writing ez=ez-1e. Homework Equations Taylor series: f(z) = Ʃ(1/2\pi/i ∫(f(z)/(z-z0)n+1dz)(z-z0)n The Attempt at a Solution The first part of this...
  30. E

    Complex variables question

    Homework Statement Show that if C is a positively oriented simple closed contour, then the area of the region enclosed by C can be written (1/2i)/∫C\bar{}zdz. Note that expression 4 Sec. 46 can be used here even though the function f(z)=\bar{}z is not analytic anywhere. FORMATTING NOTE: SHOULD...
  31. M

    Quadratic inequalities for complex variables?

    Hello, I was looking at Riley's solution manual for this specific problem. Along the way, he ended up with a quadratic inequality: If you looked at the image, he said it is given that λ is real, but he asserted that λ has no real roots because of the inequality. Doesn't that mean λ is...
  32. K

    Implicit function theorem for several complex variables

    This is the statement, in case you're not familiar with it. Let ## f_j(w,x), \; j=1, \ldots, m ## be analytic functions of ## (w,z) = (w_1, \ldots, w_m,z_1,\ldots,z_n) ## in a neighborhood of ##w^0,z^0## in ##\mathbb{C}^m \times \mathbb{C}^n ## and assume that ##f_j(w^0,z^0)=0, \...
  33. M

    Books on complex variables

    As an electrical engineering student can learning about complex variables be beneficial to me? If so can someone recommend an introductory book that I can read on my own? Also I have very little experience working with complex variables.
  34. T

    Limits at Infinity for the Argument Function in Complex Variables

    ℂI am working on an assignment and have come across a question that I'm not quite sure how to approach. Here it is, with my "solution" and reasoning: "[F]ind the limit at ∞ of the given function, or explain why it does not exist. 24. h(z) = Arg z , z \neq 0" (Complex Variables Second...
  35. micromass

    Analysis Complex Variables and Applications by Brown and Churchill

    Author: James Brown, Ruel Churchill Title: Complex Variables and Applications Amazon link https://www.amazon.com/dp/0073051942/?tag=pfamazon01-20 Table of Contents: Preface Complex Numbers Sums and Products Basic Algebraic Properties Further Properties Vectors and Moduli Complex...
  36. camilus

    Top grad programs in several complex variables and complex geometry?

    I know UC San Diego is good, Rothschild and Ebenfelt are there, but are there any other ones that stand out in these and related fields? Thanks a bunch
  37. J

    Elementary algebra of complex variables problem

    I'm having difficulty deducing that Re z = 0.
  38. M

    Complex Variables Homework: Determine Set, Compute Derivative

    Homework Statement Determine the set on which f(z) = 1/(z^3 +1) is analytic and compute its derivative. Homework Equations Hint: you do not need to appeal to the Cauchy-Riemann equations The Attempt at a Solution Total stuck with this one. everything we have done this far has...
  39. P

    Complex Analysis or Complex Variables?

    Hi everyone, I'm a Physics student going into my Junior year and I'm currently registering for my courses for the following semester and I have two options for my "complex" course, namely: --------------------------------------------------- Complex Variables Theory of functions of one complex...
  40. B

    Comp Sci Fortran Help with Complex Variables - Values Blowing Up

    Homework Statement The Fortran 90 code shown below was written to solve a Crank-Nicholson algorithm that describes the motion of a quantum particle (hence the need for complex numbers). Probably not that crucial to know the details of the math/physics, just the code... The problem is that...
  41. S

    Complex Derivative: Directional Derivatives & Complex Variables

    I'm not sure if it's OK to post this question here or not, the Calculus and Beyond section doesn't really look very heavily proof oriented. I'm trying to prove that if continuous complex valued function f(z) is such that the directional derivatives(using numbers with unit length) preserve...
  42. X

    Application of complex variables to physics?

    So I'm taking my complex variables class and learning about these cool powerful theorems like the Cauchy Goursat theorem. I know this all has huge application in physics however I just don't know what they are. Currently I'm only taking freshmen E@M so I know I won't be using it there. But next...
  43. W

    Complex Variables Limit Problem(s)

    Homework Statement a) \lim_{z\to 3i}\frac{z^2 + 9}{z - 3i} b) \lim_{z\to i}\frac{z^2 + i}{z^4 - 1} Homework Equations ? The Attempt at a Solution I'm assuming both of these are very, very similar, but I'm not quite sure how to solve them. I would like a method other than using ε...
  44. G

    Complex Variables Algebra Solutions / Argument/Modulus

    Homework Statement Solve for a, a \in \mathbb{C} \frac{2\ln(a^2 - 1)}{\pi i} = 1 Homework Equations N/A. The Attempt at a Solution Reorganizing the equation. 2\log(a^2 - 1) = \pi i
  45. C

    Integration of functions of Complex Variables

    We can show that \int_{0}^\infty e^{-kx}dx=\frac{1}{k} for real $$k>0.$$ Does this result hold for $$\Re k>0$$ belonging to complex numbers? The reason I have this question is because $$i\times\infty$$ is not $$\infty$$ and so u substitution would not work.
  46. P

    Complex Variables - Mappings

    Complex Variables - Mappings under e^z Homework Statement Find the image of S1,S2 under ez. S1 = {z=x+iy : 0 < y < \pi } S2 = {z=x+iy : x > 0, 0 < y < \pi } Homework Equations w=ez w=\rhoei\varphi \rho=ex, \varphi=y The Attempt at a Solution Did not know how to get started. I...
  47. P

    Complex Variables (Finding complex roots)

    Homework Statement Find all solutions to (z2+1)2=-1 The Attempt at a Solution I know that because it is a polynomial of degree 4 it is a square inscribed inside of a circle in the complex plane. All i really need is one solution and from that finding the other three is easy. I have tried...
  48. A

    Complex variables in compound expressions of electrodynamics

    When dealing with electrodynamics it is usual to use complex variables for the electromagnetic field while taking into account that the electromagnetic field is real and that at the end one has to take the real part of the complex solution for the field. However, what happens to compound...
  49. T

    Inverse Fourier Transform using complex variables

    Homework Statement For α > 0, determine u(x) by the inverse Fourier transform u(x) = \frac{1}{2\pi}\int_{-\infty}^{\infty}\ \frac{e^{ikx}}{ik+\alpha}\ dk Homework Equations The Attempt at a Solution This seemed like a relatively simple residue problem. You just note that...
  50. C

    Complex Variables or Stochastic Processes?

    Hi, I am a math and physics major planning on going into biophysics for grad school, and i want to do computational/mathematical modelling/theoretical work in the field. I have one more math course to take and I am not sure which would be more useful. Here are their very brief course...