cexpf, cexp, cexpl

Defined in header `<complex.h>`
float complex cexpf( float complex z );	(1)	(since C99)
double complex cexp( double complex z );	(2)	(since C99)
long double complex cexpl( long double complex z );	(3)	(since C99)
Defined in header `<tgmath.h>`
#define exp( z )	(4)	(since C99)

1-3) Computes the complex base-e exponential of z.

4) Type-generic macro: If z has type long double complex, cexpl is called. if z has type double complex, cexp is called, if z has type float complex, cexpf is called. If z is real or integer, then the macro invokes the corresponding real function (expf, exp, expl). If z is imaginary, the corresponding complex argument version is called.

Parameters

z

-

complex argument

Return value

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

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 $\small \cos(y)+{\rm i}\sin(y)$ $cos(y) + i sin(y)$

Notes

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

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

Example

Run this code

#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));
 
}

Output:

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

References

C11 standard (ISO/IEC 9899:2011):

7.3.7.1 The cexp functions (p: 194)

7.25 Type-generic math <tgmath.h> (p: 373-375)

G.6.3.1 The cexp functions (p: 543)

G.7 Type-generic math <tgmath.h> (p: 545)

C99 standard (ISO/IEC 9899:1999):

7.3.7.1 The cexp functions (p: 176)

7.22 Type-generic math <tgmath.h> (p: 335-337)

G.6.3.1 The cexp functions (p: 478)

G.7 Type-generic math <tgmath.h> (p: 480)

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Common mathematical functions
Floating-point environment (C99)
Pseudo-random number generation
Complex number arithmetic (C99)
Type-generic math (C99)

clogclogfclogl (C99)(C99)(C99)	computes the complex natural logarithm (function)
expexpfexpl (C99)(C99)	computes e raised to the given power (\({\small e^x}\) $e x$ ) (function)