# ccoshf, ccosh, ccoshl

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< c‎ | numeric‎ | complex

Defined in header `<complex.h>` |  |
float complex ccoshf( float complex z ); | (1) | (since C99)
double complex ccosh( double complex z ); | (2) | (since C99)
long double complex ccoshl( long double complex z ); | (3) | (since C99)
Defined in header `<tgmath.h>` |  |
#define cosh( z ) | (4) | (since C99)

1-3) Computes the complex hyperbolic cosine of `z`.
4) Type-generic macro: If `z` has type long double complex, `ccoshl` is called. if `z` has type double complex, `ccosh` is called, if `z` has type float complex, `ccoshf` is called. If `z` is real or integer, then the macro invokes the corresponding real function (coshf, cosh, coshl). If `z` is imaginary, then the macro invokes the corresponding real version of the function cos, implementing the formula cosh(iy) = cos(y), and the return type is real.

### Parameters

z | - | complex argument

### Return value

If no errors occur, complex hyperbolic cosine of `z` is returned

### Error handling and special values

Errors are reported consistent with math_errhandling

If the implementation supports IEEE floating-point arithmetic,

- ccosh(conj(z)) == conj(ccosh(z))
- ccosh(z) == ccosh(-z)
- If `z` is `+0+0i`, the result is `1+0i`
- If `z` is `+0+∞i`, the result is `NaN±0i` (the sign of the imaginary part is unspecified) and FE_INVALID is raised
- If `z` is `+0+NaNi`, the result is `NaN±0i` (the sign of the imaginary part is unspecified)
- If `z` is `x+∞i` (for any finite non-zero x), the result is `NaN+NaNi` and FE_INVALID is raised
- If `z` is `x+NaNi` (for any finite non-zero 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 non-zero y), the result is `+∞cis(y)`
- If `z` is `+∞+∞i`, the result is `±∞+NaNi` (the sign of the real part is unspecified) and FE_INVALID is raised
- If `z` is `+∞+NaN`, the result is `+∞+NaN`
- If `z` is `NaN+0i`, the result is `NaN±0i` (the sign of the imaginary part is unspecified)
- If `z` is `NaN+yi` (for any finite non-zero y), the result is `NaN+NaNi` and FE_INVALID may be raised
- If `z` is `NaN+NaNi`, the result is `NaN+NaNi`

where cis(y) is cos(y) + i sin(y)

### Notes

Mathematical definition of hyperbolic cosine is cosh z =

ez +e-z
2

Hyperbolic cosine is an entire function in the complex plane and has no branch cuts. It is periodic with respect to the imaginary component, with period 2πi

### Example

Run this code

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

    int main(void)
    {
        double complex z = ccosh(1);  // behaves like real cosh along the real line
        printf("cosh(1+0i) = %f%+fi (cosh(1)=%f)\n", creal(z), cimag(z), cosh(1));

        double complex z2 = ccosh(I); // behaves like real cosine along the imaginary line
        printf("cosh(0+1i) = %f%+fi ( cos(1)=%f)\n", creal(z2), cimag(z2), cos(1));
    }

Output:

    cosh(1+0i) = 1.543081+0.000000i (cosh(1)=1.543081)
    cosh(0+1i) = 0.540302+0.000000i ( cos(1)=0.540302)

### References

- C11 standard (ISO/IEC 9899:2011):

  - 7.3.6.4 The ccosh functions (p: 193)

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

  - G.6.2.4 The ccosh functions (p: 541)

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

- C99 standard (ISO/IEC 9899:1999):

  - 7.3.6.4 The ccosh functions (p: 175)

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

  - G.6.2.4 The ccosh functions (p: 476)

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

### See also

csinh csinhf csinhl (C99)(C99)(C99) | computes the complex hyperbolic sine (function)
ctanh ctanhf ctanhl (C99)(C99)(C99) | computes the complex hyperbolic tangent (function)
cacosh cacoshf cacoshl (C99)(C99)(C99) | computes the complex arc hyperbolic cosine (function)
cosh coshf coshl (C99)(C99) | computes hyperbolic cosine (ch(x)) (function)

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