# csinhf, csinh, csinhl

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< c‎ | numeric‎ | complex

Defined in header `<complex.h>` |  |
float complex csinhf( float complex z ); | (1) | (since C99)
double complex csinh( double complex z ); | (2) | (since C99)
long double complex csinhl( long double complex z ); | (3) | (since C99)
Defined in header `<tgmath.h>` |  |
#define sinh( z ) | (4) | (since C99)

1-3) Computes the complex hyperbolic sine of `z`.
4) Type-generic macro: If `z` has type long double complex, `csinhl` is called. if `z` has type double complex, `csinh` is called, if `z` has type float complex, `csinhf` is called. If `z` is real or integer, then the macro invokes the corresponding real function (sinhf, sinh, sinhl). If `z` is imaginary, then the macro invokes the corresponding real version of the function sin, implementing the formula sinh(iy) = i sin(y), and the return type is imaginary.

### Parameters

z | - | complex argument

### Return value

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

### Error handling and special values

Errors are reported consistent with math_errhandling

If the implementation supports IEEE floating-point arithmetic,

- csinh(conj(z)) == conj(csinh(z))
- csinh(z) == -csinh(-z)
- If `z` is `+0+0i`, the result is `+0+0i`
- If `z` is `+0+∞i`, the result is `±0+NaNi` (the sign of the real part is unspecified) and FE_INVALID is raised
- If `z` is `+0+NaNi`, the result is `±0+NaNi`
- If `z` is `x+∞i` (for any positive finite x), the result is `NaN+NaNi` and FE_INVALID is raised
- If `z` is `x+NaNi` (for any positive 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 positive finite 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 `+∞+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 finite nonzero 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 sine is sinh z =

ez -e-z
2

Hyperbolic sine 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 = csinh(1);  // behaves like real sinh along the real line
        printf("sinh(1+0i) = %f%+fi (sinh(1)=%f)\n", creal(z), cimag(z), sinh(1));

        double complex z2 = csinh(I); // behaves like sine along the imaginary line
        printf("sinh(0+1i) = %f%+fi ( sin(1)=%f)\n", creal(z2), cimag(z2), sin(1));
    }

Output:

    sinh(1+0i) = 1.175201+0.000000i (sinh(1)=1.175201)
    sinh(0+1i) = 0.000000+0.841471i ( sin(1)=0.841471)

### References

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

  - 7.3.6.5 The csinh functions (p: 194)

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

  - G.6.2.5 The csinh functions (p: 541-542)

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

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

  - 7.3.6.5 The csinh functions (p: 175-176)

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

  - G.6.2.5 The csinh functions (p: 476-477)

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

### See also

ccosh ccoshf ccoshl (C99)(C99)(C99) | computes the complex hyperbolic cosine (function)
ctanh ctanhf ctanhl (C99)(C99)(C99) | computes the complex hyperbolic tangent (function)
casinh casinhf casinhl (C99)(C99)(C99) | computes the complex arc hyperbolic sine (function)
sinh sinhf sinhl (C99)(C99) | computes hyperbolic sine (sh(x)) (function)

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