/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).

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

```#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)

 ccoshccoshfccoshl (C99)(C99)(C99) computes the complex hyperbolic cosine (function) ctanhctanhfctanhl (C99)(C99)(C99) computes the complex hyperbolic tangent (function) casinhcasinhfcasinhl (C99)(C99)(C99) computes the complex arc hyperbolic sine (function) sinhsinhfsinhl (C99)(C99) computes hyperbolic sine (\({\small\sinh{x} }\)sinh(x)) (function) C++ documentation for `sinh`