/C

# cacoshf, cacosh, cacoshl

float complex       cacoshf( float complex z );
(1) (since C99)
double complex      cacosh( double complex z );
(2) (since C99)
long double complex cacoshl( long double complex z );
(3) (since C99)
#define acosh( z )
(4) (since C99)
1-3) Computes complex arc hyperbolic cosine of a complex value z with branch cut at values less than 1 along the real axis.
4) Type-generic macro: If z has type long double complex, cacoshl is called. if z has type double complex, cacosh is called, if z has type float complex, cacoshf is called. If z is real or integer, then the macro invokes the corresponding real function (acoshf, acosh, acoshl). If z is imaginary, then the macro invokes the corresponding complex number version and the return type is complex.

### Parameters

 z - complex argument

### Return value

The complex arc hyperbolic cosine of z in the interval [0; ∞) along the real axis and in the interval [−iπ; +iπ] along the imaginary axis.

### Error handling and special values

Errors are reported consistent with math_errhandling.

If the implementation supports IEEE floating-point arithmetic,

• cacosh(conj(z)) == conj(cacosh(z))
• If z is ±0+0i, the result is +0+iπ/2
• If z is +x+∞i (for any finite x), the result is +∞+iπ/2
• If z is +x+NaNi (for non-zero finite x), the result is NaN+NaNi and FE_INVALID may be raised.
• If z is 0+NaNi, the result is NaN±iπ/2, where the sign of the imaginary part is unspecified
• If z is -∞+yi (for any positive finite y), the result is +∞+iπ
• If z is +∞+yi (for any positive finite y), the result is +∞+0i
• If z is -∞+∞i, the result is +∞+3iπ/4
• If z is +∞+∞i, the result is +∞+iπ/4
• If z is ±∞+NaNi, the result is +∞+NaNi
• If z is NaN+yi (for any finite y), the result is NaN+NaNi and FE_INVALID may be raised.
• If z is NaN+∞i, the result is +∞+NaNi
• If z is NaN+NaNi, the result is NaN+NaNi

### Notes

Although the C standard names this function "complex arc hyperbolic cosine", the inverse functions of the hyperbolic functions are the area functions. Their argument is the area of a hyperbolic sector, not an arc. The correct name is "complex inverse hyperbolic cosine", and, less common, "complex area hyperbolic cosine".

Inverse hyperbolic cosine is a multivalued function and requires a branch cut on the complex plane. The branch cut is conventionally placed at the line segment (-∞,+1) of the real axis.

The mathematical definition of the principal value of the inverse hyperbolic cosine is acosh z = ln(z + z+1 z-1) For any z, acosh(z) =

√z-1/√1-z acos(z), or simply i acos(z) in the upper half of the complex plane.

### Example

#include <stdio.h>
#include <complex.h>

int main(void)
{
double complex z = cacosh(0.5);
printf("cacosh(+0.5+0i) = %f%+fi\n", creal(z), cimag(z));

double complex z2 = conj(0.5); // or cacosh(CMPLX(0.5, -0.0)) in C11
printf("cacosh(+0.5-0i) (the other side of the cut) = %f%+fi\n", creal(z2), cimag(z2));

// in upper half-plane, acosh(z) = i*acos(z)
double complex z3 = casinh(1+I);
printf("casinh(1+1i) = %f%+fi\n", creal(z3), cimag(z3));
double complex z4 = I*casin(1+I);
printf("I*asin(1+1i) = %f%+fi\n", creal(z4), cimag(z4));
}

Output:

cacosh(+0.5+0i) = 0.000000-1.047198i
cacosh(+0.5-0i) (the other side of the cut) = 0.500000-0.000000i
casinh(1+1i) = 1.061275+0.666239i
I*asin(1+1i) = -1.061275+0.666239i

### References

• C11 standard (ISO/IEC 9899:2011):
• 7.3.6.1 The cacosh functions (p: 192)
• 7.25 Type-generic math <tgmath.h> (p: 373-375)
• G.6.2.1 The cacosh functions (p: 539-540)
• G.7 Type-generic math <tgmath.h> (p: 545)
• C99 standard (ISO/IEC 9899:1999):
• 7.3.6.1 The cacosh functions (p: 174)
• 7.22 Type-generic math <tgmath.h> (p: 335-337)
• G.6.2.1 The cacosh functions (p: 474-475)
• G.7 Type-generic math <tgmath.h> (p: 480)