NAME¶

std::exp(std::complex) - std::exp(std::complex)

Synopsis¶

Defined in header <complex>
template< class T >
complex<T> exp( const complex<T>& z );

Compute base-e exponential of z, that is e (Euler's number, 2.7182818) raised to the
z power.

Parameters¶

z - complex value

Return value¶

If no errors occur, e raised to the power of z, ez
, is returned.

Error handling and special values

Errors are reported consistent with math_errhandling

If the implementation supports IEEE floating-point arithmetic,

* 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 cis(y) is cos(y) + i sin(y)

Notes¶

The complex exponential function ez
for z = x+iy equals ex
cis(y), or, 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 using exp with a literal 0 component. There is no benefit in trying to
avoid exp with a runtime check of z.real() == 0 though.

Example¶

// Run this code

#include <complex>
#include <iostream>

int main()
{
const double pi = std::acos(-1);
const std::complex<double> i(0, 1);

std::cout << std::fixed << " exp(i*pi) = " << std::exp(i * pi) << '\n';
}

Output:¶

exp(i*pi) = (-1.000000,0.000000)

See also¶

complex natural logarithm with the branch cuts along the negative
log(std::complex) real axis
(function template)
exp
expf returns e raised to the given power (\({\small e^x}\)e^x)
expl (function)
(C++11)
(C++11)
exp(std::valarray) applies the function std::exp to each element of valarray
(function template)
polar constructs a complex number from magnitude and phase angle
(function template)