/C

# catanhf, catanh, catanhl

Defined in header `<complex.h>`
`float complex       catanhf( float complex z );`
(1) (since C99)
`double complex      catanh( double complex z );`
(2) (since C99)
`long double complex catanhl( long double complex z );`
(3) (since C99)
Defined in header `<tgmath.h>`
`#define atanh( z )`
(4) (since C99)
1-3) Computes the complex arc hyperbolic tangent of `z` with branch cuts outside the interval [−1; +1] along the real axis.
4) Type-generic macro: If `z` has type `long double complex`, `catanhl` is called. if `z` has type `double complex`, `catanh` is called, if `z` has type `float complex`, `catanhf` is called. If `z` is real or integer, then the macro invokes the corresponding real function (`atanhf`, `atanh`, `atanhl`). If `z` is imaginary, then the macro invokes the corresponding real version of `atan`, implementing the formula atanh(iy) = i atan(y), and the return type is imaginary.

### Parameters

 z - complex argument

### Return value

If no errors occur, the complex arc hyperbolic tangent of `z` is returned, in the range of a half-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,

• `catanh(conj(z)) == conj(catanh(z))`
• `catanh(-z) == -catanh(z)`
• If `z` is `+0+0i`, the result is `+0+0i`
• If `z` is `+0+NaNi`, the result is `+0+NaNi`
• If `z` is `+1+0i`, the result is `+∞+0i` and `FE_DIVBYZERO` is raised
• If `z` is `x+∞i` (for any finite positive x), the result is `+0+iπ/2`
• If `z` is `x+NaNi` (for any finite nonzero x), the result is `NaN+NaNi` and `FE_INVALID` may be raised
• If `z` is `+∞+yi` (for any finite positive y), the result is `+0+iπ/2`
• If `z` is `+∞+∞i`, the result is `+0+iπ/2`
• If `z` is `+∞+NaNi`, the result is `+0+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 `±0+iπ/2` (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 tangent", 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 tangent", and, less common, "complex area hyperbolic tangent".

Inverse hyperbolic tangent is a multivalued function and requires a branch cut on the complex plane. The branch cut is conventionally placed at the line segmentd (-∞,-1] and [+1,+∞) of the real axis. The mathematical definition of the principal value of the inverse hyperbolic tangent is atanh z =

 ln(1+z)-ln(z-1) 2
.

For any z, atanh(z) =

 atan(iz) i

### Example

```#include <stdio.h>
#include <complex.h>

int main(void)
{
double complex z = catanh(2);
printf("catanh(+2+0i) = %f%+fi\n", creal(z), cimag(z));

double complex z2 = catanh(conj(2)); // or catanh(CMPLX(2, -0.0)) in C11
printf("catanh(+2-0i) (the other side of the cut) = %f%+fi\n", creal(z2), cimag(z2));

// for any z, atanh(z) = atan(iz)/i
double complex z3 = catanh(1+2*I);
printf("catanh(1+2i) = %f%+fi\n", creal(z3), cimag(z3));
double complex z4 = catan((1+2*I)*I)/I;
printf("catan(i * (1+2i))/i = %f%+fi\n", creal(z4), cimag(z4));
}```

Output:

```catanh(+2+0i) = 0.549306+1.570796i
catanh(+2-0i) (the other side of the cut) = 0.549306-1.570796i
catanh(1+2i) = 0.173287+1.178097i
catan(i * (1+2i))/i = 0.173287+1.178097i```
• C11 standard (ISO/IEC 9899:2011):
• 7.3.6.3 The catanh functions (p: 193)
• 7.25 Type-generic math <tgmath.h> (p: 373-375)
• G.6.2.3 The catanh functions (p: 540-541)
• G.7 Type-generic math <tgmath.h> (p: 545)
• C99 standard (ISO/IEC 9899:1999):
• 7.3.6.3 The catanh functions (p: 175)
• 7.22 Type-generic math <tgmath.h> (p: 335-337)
• G.6.2.3 The catanh functions (p: 475-476)
• G.7 Type-generic math <tgmath.h> (p: 480)