Complex Exponentiation Operation
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A Complex Exponentiation Operation is an exponentiation operation [math]\displaystyle{ A^B }[/math] where A and B are complex numbers.
- Context:
- It can be represented [math]\displaystyle{ (x+iy)^{(a+ib)} }[/math] where [math]\displaystyle{ a, b, x, y }[/math] are real numbers.
- It can ranges from being a Real Number to a Complex Number.
- It can (typically) have the property that [math]\displaystyle{ z^w=(|z|e^{iarg(z)})^w=|z|^we^{iwarg(w)} }[/math] with [math]\displaystyle{ |z| }[/math] denoting the complex modulus and [math]\displaystyle{ arg(z) }[/math] its argument.
- For any two complex numbers [math]\displaystyle{ z=a+ib }[/math] and [math]\displaystyle{ w=u+iv }[/math], it is expressed:
[math]\displaystyle{ (a+ib)^{(u+iv)} = (a^2+b^2)^{u/2}e^{-varg(a+ib)}(cos\theta+isin\theta) }[/math] with [math]\displaystyle{ \theta=uarg(a+ib)+\frac{v}{2}ln(a^2+b^2) }[/math] where [math]\displaystyle{ ln }[/math] is the natural logarithm function and [math]\displaystyle{ arg }[/math] the complex number argument
- Example(s):
- [math]\displaystyle{ (i+1)^{(i+1)}=\sqrt{2}e^{-\pi/2}(0.425+i0.905) }[/math]
- [math]\displaystyle{ i^i= e^{-\pi/2} }[/math]
- Counter-Example(s):
- [math]\displaystyle{ 2^3=8 }[/math]
- [math]\displaystyle{ e^2=7.3891 }[/math]
- See: Complex Number, Complex Exponential Function.