Complex variables Definition and 11 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. 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...
  2. 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
  3. 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...
  4. 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...
  5. 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...
  6. 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...
  7. H

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

    Is this function analytic or not? Please explain
  8. 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|...
  9. 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...
  10. V

    Topological implications of symbolic represenation of the relativity

    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...
  11. 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...