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csinhf, csinh, csinhl

From cppreference.com
< 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.

Contents

[edit] Parameters

z - complex argument

[edit] Return value

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

[edit] 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)

[edit] 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

[edit] 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)

[edit] See also

(C99)(C99)(C99)
computes the complex hyperbolic cosine
(function) [edit]
(C99)(C99)(C99)
computes the complex hyperbolic tangent
(function) [edit]
(C99)(C99)(C99)
computes the complex arc hyperbolic sine
(function) [edit]
(C99)(C99)
computes hyperbolic sine (sh(x))
(function) [edit]