csinhf, csinh, csinhl

 

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Complex number arithmetic

Types and the imaginary constant
complex _Complex_I CMPLX	imaginary _Imaginary_I I
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cimag creal carg	cabs conj cproj
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cexp clog	cpow csqrt
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ccos csin ctan	cacos casin catan
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ccosh csinh ctanh	cacosh casinh catanh

 

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)

See also

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)
C++ documentation for sinh