C library - catanh() function

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The C complex library catanh() function is used to calculate the complex arc hyperbolic tangent, i.e. inverse hyperbolic tangent of z with branch cuts that is lie outside the interval [−1,+1] along the real axis. It makes the function continuous and single-valued for any complex number z not lying in this interval on the real axis.

The arc hyperbolic tangent (atanh) z is defined as:
atanh(z) = 1/2ln(1+z/1-z)

This function is depends on the type of z. If the z is the "float" type, we use catanhf() to compute arc hyperbolic tangent, For long double type, use catanhl(), and for double type, use catanh().

Syntax

Following is the C library syntax of catanh() function −

double complex catanh( double complex z );

Parameters

This function accepts a single parameter −

• Z − It represent a complex number for which we want to calculate atanh.

Return Value

This function returns the complex inverse (arc) hyperbolic tangent of z within a half-strip that is unbounded along the real axis, the imaginary axis always lying between [−iπ/2, +iπ/2].

Example 1

Following is the basic c program to demonstrate the use of catanh() on a complex number.

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

int main() {
   // Define z
   double complex z = 0.5 + 0.5*I;

   // Calculate the complex inverse hyperbolic tangent of z
   double complex res= catanh(z);

   // Display the result
   printf("catanh(%.2f + %.2fi) = %.2f + %.2fi\n", creal(z), cimag(z), creal(res), cimag(res));

   return 0;
}

Output

Following is the output −

catanh(0.50 + 0.50i) = 0.40 + 0.55i

Example 2

Let's see another example, calculate the inverse hyperbolic tangent of real axis using catanh() function.

#include <stdio.h>
#include <math.h>
#include <complex.h>
 
int main(void)
{
   // real axis
   double complex z = catanh(1);
   printf("tanh(1+0i) = %.2f+%.2fi \n", creal(z), cimag(z));
}

Output

Following is the output −

tanh(1+0i) = inf+0.00i

Example 3

The below program, calculates both arc hyperbolic tangent(atanh) and hyperbolic tangent(tanh) of the imaginary line of a complex number, and then compares the answer to see if they are same.

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

int main() {
   double complex z = 0.0 + 1.0*I;

   double complex atanh = catanh(z);
   double complex tanh = ctanh(z);

   printf("catanh(%.1fi) = %.2f + %.2fi\n", cimag(z), creal(atanh), cimag(atanh));
   printf("ctanh(%.1fi) = %.2f + %.2fi\n", cimag(z), creal(tanh), cimag(tanh));

   if (cabs(atanh) == cabs(tanh)) {
      printf("The arc hyperbolic tangent and hyperbolic tangent of the imaginary line are approximately the same.\n");
   } else {
      printf("The arc hyperbolic tangent and hyperbolic tangent of the imaginary line are different.\n");
   }
   return 0;
}

Output

Following is the output −

catanh(1.0i) = 0.00 + 1.56i
ctan(1.0i) = 0.00 + 0.76i
The hyperbolic tangent and tangent of the imaginary line are different.