Get Started »

# Complex Analysis

The imaginary unit is represented as I:

 In[1]:= ⨯ `I^2`
 Out[1]=

Most operations automatically handle complex numbers:

 In[2]:= ⨯ `(1 + I) (2 - 3 I)`
 Out[2]=

Expand complex expressions:

 In[1]:= ⨯ `ComplexExpand[Sin[x + I y]]`
 Out[1]=

Convert expressions between exponential and trig forms:

 In[2]:= ⨯ `ExpToTrig[E^(I x)]`
 Out[2]=
 In[3]:= ⨯ `TrigToExp[%]`
 Out[3]=

Type ESCcoESC for the Conjugate symbol:

 In[1]:= ⨯ `(3 + 2 I)\[Conjugate]`
 Out[1]=

Extract the real and imaginary parts of an expression:

 In[2]:= ⨯ `ReIm[3 + 2 I]`
 Out[2]=

Or find the absolute value and argument:

 In[3]:= ⨯ `AbsArg[(1 + I)]`
 Out[3]=

Plot a conformal mapping with ParametricPlot:

 In[1]:= ⨯ `ParametricPlot[ReIm[E^(I \[Omega])], {\[Omega], 0, 2 \[Pi]}]`
 Out[1]=

Use AbsArg in a PolarPlot:

 In[2]:= ⨯ `PolarPlot[AbsArg[E^(I \[Omega])], {\[Omega], 0, \[Pi]}]`
 Out[2]=

Visualize complex components with a DensityPlot:

 In[3]:= ⨯ ```DensityPlot[Im[ArcSin[(x + I y)^2]], {x, -2, 2}, {y, -2, 2}]```
 Out[3]=

QUICK REFERENCE: Complex Numbers `»`

QUICK REFERENCE: Functions of Complex Variables `»`