# catanhf, catanh, catanhl

< 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       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  #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.

## Contents

### 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

### Notes

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

### References

• 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)