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Showing content from https://en.cppreference.com/w/cpp/language/../algorithm/../../cpp/../c/numeric/complex/catanh.html below:

catanhf, catanh, catanhl - cppreference.com

#define atanh( z )

1-3) Computes the complex arc hyperbolic tangent of z with branch cuts outside the interval [−1; +1] along the real axis.

Type-generic macro: If

has type

,

is called. if

has type

,

is called, if

has type

,

is called. If

is real or integer, then the macro invokes the corresponding real function (

,

,

). If

is imaginary, then the macro invokes the corresponding real version of

, implementing the formula

, and the return type is imaginary.

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.

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

.

For any 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));
}

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)

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