cacoshf, cacosh, cacoshl

Defined in header `<complex.h>`
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)
Defined in header `<tgmath.h>`
#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 $[0; \infty)$ 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 any^[1] finite x), the result is NaN+NaNi and FE_INVALID may be raised.
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 ±∞+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

↑ per DR471, this holds for non-zero x only. If z is 0+NaNi, the result should be NaN+iπ/2

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 $(-\infty,+1)$ of the real axis.

The mathematical definition of the principal value of the inverse hyperbolic sine is $acosh z = ln(z + \sqrt z+1 + \sqrt z-1)$

For any z,

acosh(z) = \sqrt z-1 \sqrt 1-z acos(z)

, or simply

i acos(z)

in the upper half of the complex plane.

Example

Run this code

#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)

Language
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Type support
Dynamic memory management
Error handling
Program utilities
Variadic function support
Date and time utilities
Strings library
Algorithms
Numerics
Input/output support
Localization support
Thread support (C11)
Atomic operations (C11)
Technical Specifications

Common mathematical functions
Floating-point environment
Complex number arithmetics
Random number generation
Type-generic math

cacoscacosfcacosl (C99)(C99)(C99)	computes the complex arc cosine (function)
casinhcasinhfcasinhl (C99)(C99)(C99)	computes the complex arc hyperbolic sine (function)
catanhcatanhfcatanhl (C99)(C99)(C99)	computes the complex arc hyperbolic tangent (function)
ccoshccoshfccoshl (C99)(C99)(C99)	computes the complex hyperbolic cosine (function)
acoshacoshfacoshl (C99)(C99)(C99)	computes inverse hyperbolic cosine ( $arcosh(x)$ ) (function)
C++ documentation for acosh