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casinhf, casinh, casinhl

From cppreference.com
< c‎ | numeric‎ | complex
Defined in header <complex.h>
float complex       casinhf( float complex z );
(1) (since C99)
double complex      casinh( double complex z );
(2) (since C99)
long double complex casinhl( long double complex z );
(3) (since C99)
Defined in header <tgmath.h>
#define asinh( z )
(4) (since C99)
1-3) Computes the complex arc hyperbolic sine of z with branch cuts outside the interval [−i; +i] along the imaginary axis.
4) Type-generic macro: If z has type long double complex, casinhl is called. if z has type double complex, casinh is called, if z has type float complex, casinhf is called. If z is real or integer, then the macro invokes the corresponding real function (asinhf, asinh, asinhl). If z is imaginary, then the macro invokes the corresponding real version of the function asin, implementing the formula asinh(iy) = i asin(y), and the return type is imaginary.

Contents

[edit] Parameters

z - complex argument

[edit] Return value

If no errors occur, the complex arc hyperbolic sine of z is returned, in the range of a strip mathematically unbounded along the real axis and in the interval [−iπ/2; +iπ/2] along the imaginary axis.

[edit] Error handling and special values

Errors are reported consistent with math_errhandling

If the implementation supports IEEE floating-point arithmetic,

  • casinh(conj(z)) == conj(casinh(z))
  • casinh(-z) == -casinh(z)
  • If z is +0+0i, the result is +0+0i
  • If z is x+∞i (for any positive finite x), the result is +∞+π/2
  • If z is x+NaNi (for any finite x), the result is NaN+NaNi and FE_INVALID may be raised
  • If z is +∞+yi (for any positive finite y), the result is +∞+0i
  • If z is +∞+∞i, the result is +∞+iπ/4
  • If z is +∞+NaNi, the result is +∞+NaNi
  • 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+∞i, the result is ±∞+NaNi (the sign of the real part is unspecified)
  • If z is NaN+NaNi, the result is NaN+NaNi

[edit] Notes

Although the C standard names this function "complex arc hyperbolic sine", the inverse functions of the hyperbolic functions are the area functions. Their argument is the area of a hyperbolic sector, not an arc. The correct name is "complex inverse hyperbolic sine", and, less common, "complex area hyperbolic sine".

Inverse hyperbolic sine is a multivalued function and requires a branch cut on the complex plane. The branch cut is conventionally placed at the line segments (-i∞,-i) and (i,i∞) of the imaginary axis.

The mathematical definition of the principal value of the inverse hyperbolic sine is asinh z = ln(z + 1+z2
)

For any z, asinh(z) =
asin(iz)
i

[edit] Example

#include <stdio.h>
#include <complex.h>
 
int main(void)
{
    double complex z = casinh(0+2*I);
    printf("casinh(+0+2i) = %f%+fi\n", creal(z), cimag(z));
 
    double complex z2 = casinh(-conj(2*I)); // or casinh(CMPLX(-0.0, 2)) in C11
    printf("casinh(-0+2i) (the other side of the cut) = %f%+fi\n", creal(z2), cimag(z2));
 
    // for any z, asinh(z) = asin(iz)/i
    double complex z3 = casinh(1+2*I);
    printf("casinh(1+2i) = %f%+fi\n", creal(z3), cimag(z3));
    double complex z4 = casin((1+2*I)*I)/I;
    printf("casin(i * (1+2i))/i = %f%+fi\n", creal(z4), cimag(z4));
}

Output:

casinh(+0+2i) = 1.316958+1.570796i
casinh(-0+2i) (the other side of the cut) = -1.316958+1.570796i
casinh(1+2i) = 1.469352+1.063440i
casin(i * (1+2i))/i =  1.469352+1.063440i

[edit] See also

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