ComplexExpand

Split expr into real and imaginary parts, treating every free symbol as real. Provided by @enumeratio/analytic.

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ComplexExpand(expr)expr, rewritten as Re + i·Im with its symbols assumed real.

Typed signatures

Type	Package	Applies
`(expression) -> expression`	`analytic`	always

Domain: Transformations

Details

Covers Add, Multiply, Negate, a nonnegative-integer Power, Sin, Cos, Sinh, Cosh, Exp and Abs of a complex argument, a power of a concrete complex base (I^x is $e^{i\pi x/2}$ ), the real and imaginary parts of a split, and a list entry by entry. Tan, Tanh, Ln and the other heads have no rule and fall through to "assumed real", which is wrong for them specifically.
A concrete numeric argument with no free symbols needs none of this: plain evaluation already gives the same split (see the last example), so this is a no-op there.

Examples

Basic #

$\sin(x + iy) = \sin x\cosh y + i\cos x\sinh y$

In

\mathrm{ComplexExpand}(\sin(x+i y))

Out

i\cos(x)\sinh(y)+\sin(x)\cosh(y)

Polar form of a complex exponential

In

\mathrm{ComplexExpand}(\exp(x+i y))

Out

i\sin(y)e^{x}+\cos(y)e^{x}

In

\mathrm{ComplexExpand}(\mathrm{ReIm}(\cos(i^{x})))

Out

\bigl\lbrack\cos(\cos(\frac{πx}{2}))\cosh(\sin(\frac{πx}{2})), -(\sin(\cos((πx)/2))\sinh(\sin((πx)/2)))\bigr\rbrack

Scope #

In

\mathrm{ComplexExpand}((x+i y)^2)

Out

x^2-y^2+2i xy

In

\mathrm{ComplexExpand}(\vert x+i y\vert)

Out

\sqrt{x^2+y^2}

Wolfram leaves $e^{i\pi/5}$ alone until asked; this is the radical form compute-engine's Exp already produces

In

\mathrm{ComplexExpand}(\exp(\frac{i π}{5}))

Out

\frac{1}{4}+\frac{\sqrt{5}}{4}+\frac{\sqrt{2}}{4}i\sqrt{5-\sqrt{5}}

In

\mathrm{ComplexExpand}(\Re(\ln(\sin(\exp(2(x+i y))))))

Out

\frac{1}{2}(\ln(\sin(\cos(2y)\exp(2x))^2\cosh(\sin(2y)\exp(2x))^2+\cos(\cos(2y)\exp(2x))^2\sinh(\sin(2y)\exp(2x))^2))

Properties #

In

\mathrm{ComplexExpand}(\Re(\sin(x+i y)))

Out

\sin(x)\cosh(y)

About

Examples	8
Checked against	Wolfram Language 8 agree
Crosswalk	Wolfram Language

See also: ExpToTrig, Re, Im