/C

# csinhf, csinh, csinhl

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).

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

```#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)```
• 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)