Wolfram Language Fast Introduction for Math Students
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Complex Analysis

The imaginary unit is represented as I:

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I^2
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Most operations automatically handle complex numbers:

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⨯ 
(1 + I) (2 - 3 I)
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Expand complex expressions:

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⨯ 
ComplexExpand[Sin[x + I y]]
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Convert expressions between exponential and trig forms:

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⨯ 
ExpToTrig[E^(I x)]
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⨯ 
TrigToExp[%]
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Type ESCcoESC for the Conjugate symbol:

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⨯ 
(3 + 2 I)\[Conjugate]
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Extract the real and imaginary parts of an expression:

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⨯ 
ReIm[3 + 2 I]
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Or find the absolute value and argument:

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⨯ 
AbsArg[(1 + I)]
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Plot a conformal mapping with ParametricPlot:

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ParametricPlot[ReIm[E^(I \[Omega])], {\[Omega], 0, 2 \[Pi]}]
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Use AbsArg in a PolarPlot:

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PolarPlot[AbsArg[E^(I \[Omega])], {\[Omega], 0, \[Pi]}]
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Visualize complex components with a DensityPlot:

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⨯ 
DensityPlot[Im[ArcSin[(x + I y)^2]],
 {x, -2, 2}, {y, -2, 2}]
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QUICK REFERENCE: Complex Numbers »

QUICK REFERENCE: Functions of Complex Variables »