Compute base-e exponential ofz, that ise (Euler's number,2.7182818) raised to thez power.

[edit]Parameters

[edit]Return value

If no errors occur,e raised to the power ofz,\(\small e^z\)ez
, is returned.

[edit]Error handling and special values

Errors are reported consistent withmath_errhandling.

If the implementation supports IEEE floating-point arithmetic,

• std::exp(std::conj(z))==std::conj(std::exp(z))
• Ifz is(±0,+0), the result is(1,+0)
• Ifz is(x,+∞) (for any finite x), the result is(NaN,NaN) andFE_INVALID is raised.
• Ifz is(x,NaN) (for any finite x), the result is(NaN,NaN) andFE_INVALID may be raised.
• Ifz is(+∞,+0), the result is(+∞,+0)
• Ifz is(-∞,y) (for any finite y), the result is+0cis(y)
• Ifz is(+∞,y) (for any finite nonzero y), the result is+∞cis(y)
• Ifz is(-∞,+∞), the result is(±0,±0) (signs are unspecified)
• Ifz is(+∞,+∞), the result is(±∞,NaN) andFE_INVALID is raised (the sign of the real part is unspecified)
• Ifz is(-∞,NaN), the result is(±0,±0) (signs are unspecified)
• Ifz is(+∞,NaN), the result is(±∞,NaN) (the sign of the real part is unspecified)
• Ifz is(NaN,+0), the result is(NaN,+0)
• Ifz is(NaN,y) (for any nonzero y), the result is(NaN,NaN) andFE_INVALID may be raised
• Ifz 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 anentire 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 caseexp can be about 4.5x slower. One of the other forms should be used instead of callingexp with an argument whose real part is literal 0. There is no benefit in trying to avoidexp with a runtime check ofz.real()==0 though.

[edit]Example

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

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

[edit]See also

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