Examples for

Complex Analysis

Complex analysis is the field of mathematics dealing with the study of complex numbers and functions of a complex variable. Wolfram|Alpha's authoritative computational ability allows you to perform complex arithmetic, analyze and compute properties of complex functions and apply the methods of complex analysis to solve related mathematical queries.

Complex Numbers ›

Analyze properties of functions of a complex variable or perform basic arithmetic on, find roots of or apply functions to complex numbers.

Do basic arithmetic on complex numbers:

((3+4i)/5)^10

Apply functions to complex numbers:

exp(24+2i)
More examples

Poles

Locate poles of a complex function within a specified domain or within the entire complex plane.

Locate poles of a complex function:

poles of (z^2-4) / ((z-2)^4*(z^2+5z+7))
pole points of 1/(sin^2 z)
show poles of Gamma(z)

Locate poles in a specified domain:

poles of z tan(iz) with 0 < Im z < 12

Complex Functions

Plot functions of a complex variable or compute and analyze their properties.

Compute properties of a function of a complex variable (use the variable z):

e^z
sqrt(z)

Plot a complex function:

plot z^4

Construct an analytic function:

analytic function with real part x^2 - y^2
holomorphic function imaginary part Sinh[x] Sin[y]

Determine the harmonic conjugate:

harmonic conjugate x^3 - 3 x y^2

Residues

Compute the residues of functions in the complex plane at a point or within a specified region.

Compute the residue of a function at a point:

residue of 1/(z^2+4)^2 at z=2i
csc^7(z) residue at z=0

Compute residues at the poles of a function:

show residues (z^2-9)/(z^3-3z^2+2z)

Compute residues at poles in a specified domain:

residues of 1/(e^(2z)+1) with |z| < 3pi

RELATED EXAMPLES

  • Calculus & Analysis
  • Differential Equations
  • Quaternions

    • RELATED WOLFRAM RESOURCES

  • Explore Complex Analysis at Wolfram MathWorld
  • Wolfram Complex Numbers Guide Page
  • Wolfram Functions of Complex Variables Guide Page
  • Wolfram Tutorial on Expressions Involving Complex Variables

    • Riemann Surfaces

    Compute and visualize Riemann surfaces for complex functions.

    Visualize a Riemann surface:

    Riemann surface log z
    Riemann surface (z^3-z)^(1/3)