Exponentiation of complex numbers

Complex exponentiation of a complex number

Complex exponentiation of a complex number

SeedRandom[9999]

z=RandomComplex[]

0.574500830862519` +0.4272486538050493` Ic

exponent=RandomComplex[]

(z^exponent)

(z^exponent)==Exp[exponent*Log[z]]

((Conjugate[z])^exponent)==Exp[exponent*Conjugate[Log[z]]]
((Conjugate[z])^exponent)==Exp[exponent*Log[Conjugate[z]]]

Purely real exponentiation of a complex number

SeedRandom[9999]

z=RandomReal[1]+RandomReal[1]*I

exponent=Re[RandomComplex[]]

(z^exponent)

(z^exponent)==Exp[exponent*Log[z]]

((Conjugate[z])^exponent)==Exp[exponent*Conjugate[Log[z]]]
((Conjugate[z])^exponent)==Exp[exponent*Log[Conjugate[z]]]

Conjugate[Log[z]]==Log[Conjugate[z]]

Purely imaginary exponentiation of a complex number

SeedRandom[9999]

z=RandomReal[1]+RandomReal[1]*I

exponent=Im[RandomComplex[]]

(z^exponent)

(z^exponent)==Exp[exponent*Log[z]]

((Conjugate[z])^exponent)==Exp[exponent*Conjugate[Log[z]]]
((Conjugate[z])^exponent)==Exp[exponent*Log[Conjugate[z]]]