std::exp(std::complex)

std::exp (std::conj (z)) == std::conj (std::exp (z))
If z is (±0,+0), the result is (1,+0)
If z is (x,+∞) (for any finite x), the result is (NaN,NaN) and FE_INVALID is raised.
If z is (x,NaN) (for any finite x), the result is (NaN,NaN) and FE_INVALID may be raised.
If z is (+∞,+0), the result is (+∞,+0)
If z is (-∞,y) (for any finite y), the result is +0cis(y)
If z is (+∞,y) (for any finite nonzero y), the result is +∞cis(y)
If z is (-∞,+∞), the result is (±0,±0) (signs are unspecified)
If z is (+∞,+∞), the result is (±∞,NaN) and FE_INVALID is raised (the sign of the real part is unspecified)
If z is (-∞,NaN), the result is (±0,±0) (signs are unspecified)
If z is (+∞,NaN), the result is (±∞,NaN) (the sign of the real part is unspecified)
If z is (NaN,+0), the result is (NaN,+0)
If z is (NaN,y) (for any nonzero y), the result is (NaN,NaN) and FE_INVALID may be raised
If z is (NaN,NaN), the result is (NaN,NaN)

where \(\small{\rm cis}(y)\)cis(y) is \(\small \cos(y)+{\rm i}\sin(y)\)cos(y) + i sin(y).

[edit] Notes

The complex exponential function \(\small e^z\)ez
for \(\small z = x + {\rm i}y\)z = x+iy equals \(\small e^x {\rm cis}(y)\)ex
cis(y), or, \(\small e^x (\cos(y)+{\rm i}\sin(y))\)ex
(cos(y) + i sin(y)).

The exponential function is an entire function in the complex plane and has no branch cuts.

The following have equivalent results when the real part is 0:

std::exp (std::complex <float>(0, theta))
std::complex <float>(cosf(theta), sinf(theta))
std::polar (1.f, theta)

In this case exp can be about 4.5x slower. One of the other forms should be used instead of calling exp with an argument whose real part is literal 0. There is no benefit in trying to avoid exp with a runtime check of z.real() == 0 though.

[edit] Example

Run this code

#include <cmath>
#include <complex>
#include <iostream>
 
int main()
{
 const double pi = std::acos (-1.0);
 const std::complex <double> i(0.0, 1.0);
 
 std::cout << std::fixed << " exp(i * pi) = " << std::exp (i * pi) << '\n';
}