# cacoshf, cacosh, cacoshl

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< c‎ | numeric‎ | complex

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; ∞) 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 `±∞+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

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)

### See also

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)

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