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 = e z -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)

Language
Headers
Type support
Program utilities
Variadic function support
Error handling
Dynamic memory management
Date and time utilities
Strings library
Algorithms
Numerics
Input/output support
Localization support
Atomic operations (C11)
Thread support (C11)
Technical Specifications

Common mathematical functions
Floating-point environment
Complex number arithmetics
Random number generation
Type-generic math

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 ( $sh(x)$ ) (function)