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Showing content from https://en.cppreference.com/w/cpp/language/../algorithm/../../cpp/../c/numeric/complex/casinh.html below:

casinhf, casinh, casinhl - cppreference.com

#define asinh( z )

1-3) Computes the complex arc hyperbolic sine of z with branch cuts outside the interval [−i; +i] along the imaginary axis.

Type-generic macro: If

has type

,

is called. if

has type

,

is called, if

has type

,

is called. If

is real or integer, then the macro invokes the corresponding real function (

,

,

). If

is imaginary, then the macro invokes the corresponding real version of the function

, implementing the formula

, and the return type is imaginary.

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.

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

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,

#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

• C11 standard (ISO/IEC 9899:2011):

• 7.3.6.2 The casinh functions (p: 192-193)

• 7.25 Type-generic math <tgmath.h> (p: 373-375)

• G.6.2.2 The casinh functions (p: 540)

• G.7 Type-generic math <tgmath.h> (p: 545)

• C99 standard (ISO/IEC 9899:1999):

• 7.3.6.2 The casinh functions (p: 174-175)

• 7.22 Type-generic math <tgmath.h> (p: 335-337)

• G.6.2.2 The casinh functions (p: 475)

• G.7 Type-generic math <tgmath.h> (p: 480)

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