NAME¶

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

Synopsis¶

Defined in header <complex>
template< class T >
std::complex<T> exp( const std::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, \(\small e^z\)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 \(\small{\rm cis}(y)\)cis(y) is \(\small \cos(y)+{\rm i}\sin(y)\)cos(y) + i
sin(y).

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.

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';
}

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)
C documentation for
cexp