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cacoshf, cacosh, cacoshl

From cppreference.com
< c‎ | numeric‎ | complex
Defined in header <complex.h>
float complex       cacoshf( float complex z );
(1) (since C99)
double complex      cacosh( double complex z );
(2) (since C99)
long double complex cacoshl( long double complex z );
(3) (since C99)
Defined in header <tgmath.h>
#define acosh( z )
(4) (since C99)
1-3) Computes complex arc hyperbolic sine of a complex value z with branch cut at values less than 1 along the real axis.
4) Type-generic macro: If z has type long double complex, cacoshl is called. if z has type double complex, cacosh is called, if z has type float complex, cacoshf is called. If z is real or integer, then the macro invokes the corresponding real function (acoshf, acosh, acoshl). If z is imaginary, then the macro invokes the corresponding complex number version and the return type is complex.

Contents

[edit] Parameters

z - complex argument

[edit] Return value

The complex arc hyperbolic cosine of z in the interval [0; ∞) along the real axis and in the interval [−iπ; +iπ] 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,

  • cacosh(conj(z)) == conj(cacosh(z))
  • If z is ±0+0i, the result is +0+iπ/2
  • If z is +x+∞i (for any finite x), the result is +∞+iπ/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 +∞+iπ
  • If z is +∞+yi (for any positive finite y), the result is +∞+0i
  • If z is -∞+∞i, the result is +∞+3iπ/4
  • If z is ±∞+NaNi, the result is +∞+NaNi
  • If z is NaN+yi (for any finite y), the result is NaN+NaNi and FE_INVALID may be raised.
  • If z is NaN+∞i, the result is +∞+NaNi
  • If z is NaN+NaNi, the result is NaN+NaNi

[edit] Notes

Although the C standard names this function "complex arc hyperbolic cosine", 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 cosine", and, less common, "complex area hyperbolic cosine".

Inverse hyperbolic cosine is a multivalued function and requires a branch cut on the complex plane. The branch cut is conventionally placed at the line segment (-∞,+1) of the real axis.

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

For any z, acosh(z) =
z-1
1-z
acos(z)
, or simply i acos(z) in the upper half of the complex plane.

[edit] Example

#include <stdio.h>
#include <complex.h>
 
int main(void)
{
    double complex z = cacosh(0.5);
    printf("cacosh(+0.5+0i) = %f%+fi\n", creal(z), cimag(z));
 
    double complex z2 = conj(0.5); // or cacosh(CMPLX(0.5, -0.0)) in C11
    printf("cacosh(+0.5-0i) (the other side of the cut) = %f%+fi\n", creal(z2), cimag(z2));
 
    // in upper half-plane, acosh(z) = i*acos(z) 
    double complex z3 = casinh(1+I);
    printf("casinh(1+1i) = %f%+fi\n", creal(z3), cimag(z3));
    double complex z4 = I*casin(1+I);
    printf("I*asin(1+1i) = %f%+fi\n", creal(z4), cimag(z4));
}

Output:

cacosh(+0.5+0i) = 0.000000-1.047198i
cacosh(+0.5-0i) (the other side of the cut) = 0.500000-0.000000i
casinh(1+1i) = 1.061275+0.666239i
I*asin(1+1i) = -1.061275+0.666239i

[edit] See also

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