# casinhf, casinh, casinhl

< c‎ | numeric‎ | complex

C
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Complex number arithmetic
Types and the imaginary constant
Manipulation
Power and exponential functions
Trigonometric functions
Hyperbolic functions
 cacosh casinh catanh

 Defined in header  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  #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

### Parameters

 z - complex argument

### 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.

### 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

### 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

### 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

### References

• 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)