A complex number may be taken to the power of another complex number. In particular, complex exponentiation satisfies (a + b i)^(c + d i) = (a^2 + b^2)^((c + i d)/2) e^(i(c + i d) arg(a + i b)), where arg(z) is the complex argument. Written explicitly in terms of real and imaginary parts, (a + b i)^(c + d i) = (a^2 + b^2)^(c/2) e^(-d arg(a + i b))×{cos[c arg(a + i b) + 1/2 d ln(a^2 + b^2)] + i sin[c arg(a + i b) + 1/2 d ln(a^2 + b^2)]}. An explicit example of complex exponentiation is given by (1 + i)^(1 + i) = sqrt(2)e^(-π/4)[cos(1/4 π + 1/2 ln2) + i sin(1/4 π + 1/2 ln2)].

complex addition | complex division | complex multiplication | complex number | complex subtraction | exponent | exponential function | exponent laws | power