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.


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 =


For any z, atanh(z) =



#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));


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):
    • 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):
    • 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

computes the complex arc hyperbolic sine
computes the complex arc hyperbolic cosine
computes the complex hyperbolic tangent
computes inverse hyperbolic tangent (\({\small\operatorname{artanh}{x} }\)artanh(x))
C++ documentation for atanh

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