cexpf, cexp, cexpl

[edit] Parameters

[edit] Return value

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

[edit] Error handling and special values

Errors are reported consistent with math_errhandling.

If the implementation supports IEEE floating-point arithmetic,

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

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.

[edit] Example

#include <stdio.h>
#include <math.h>
#include <complex.h>
 
int main(void)
{
 double PI = acos (-1);
 double complex z = cexp(I * PI); // Euler's formula
 printf ("exp(i*pi) = %.1f%+.1fi\n", creal (z), cimag (z));
 
}

exp(i*pi) = -1.0+0.0i

[edit] References

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