/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)

### See also

 casinhcasinhfcasinhl (C99)(C99)(C99) computes the complex arc hyperbolic sine (function) cacoshcacoshfcacoshl (C99)(C99)(C99) computes the complex arc hyperbolic cosine (function) ctanhctanhfctanhl (C99)(C99)(C99) computes the complex hyperbolic tangent (function) atanhatanhfatanhl (C99)(C99)(C99) computes inverse hyperbolic tangent (artanh(x)) (function)

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