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.
More examples
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
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
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
More examples
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
More examples
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)
More examples