// Override.
// (r * exp(i * theta))^(a + bi) = r^(a + bi) * exp(i * theta * a - theta * b)
// = (r^a * exp(-theta * b)) * (r^b * exp(theta * a))^i
ComplexNumber Power::eval(const ComplexNumber z) const {
double r = z.mag(), theta = z.angle();
double a = exponent.real(), b = exponent.imag();
double coeff = pow(r, a) * exp(-theta * b);
ComplexNumber w = iPower(pow(r, b) * exp(theta * a));
ComplexNumber v (coeff * w.real(), coeff * w.imag());
return v;
}
// Override.
// ln(z) = ln(|r|) + i * theta, where z = r * e^(i * theta)
// log_z(w) = ln(w) * log_z(e) = ln(w) * 1 / (ln(r) + i * theta), where z = r * e^(i * theta).
ComplexNumber Logarithm::eval(const ComplexNumber z) const {
ComplexNumber argument = z + lambda;
ComplexNumber ln_z (log(argument.mag()), argument.angle());
if (base == exp(1)) {
return coefficient * ln_z;
} else {
ComplexNumber one (1, 0);
ComplexNumber log_base (log(base.mag()), base.angle());
return coefficient * ln_z * (one / log_base);
}
}
