/C

# casinhf, casinh, casinhl

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.

### 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)

 cacoshcacoshfcacoshl (C99)(C99)(C99) computes the complex arc hyperbolic cosine (function) catanhcatanhfcatanhl (C99)(C99)(C99) computes the complex arc hyperbolic tangent (function) csinhcsinhfcsinhl (C99)(C99)(C99) computes the complex hyperbolic sine (function) asinhasinhfasinhl (C99)(C99)(C99) computes inverse hyperbolic sine (\({\small\operatorname{arsinh}{x} }\)arsinh(x)) (function) C++ documentation for `asinh`