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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,

-

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

See also

- - - - -
-
(C99)(C99)(C99)
 computes the complex arc hyperbolic sine
(function)
-
(C99)(C99)(C99)
 computes the complex arc hyperbolic cosine
(function)
-
(C99)(C99)(C99)
 computes the complex hyperbolic tangent
(function)
-
(C99)(C99)(C99)
 computes inverse hyperbolic tangent (\({\small\operatorname{artanh}{x} }\)artanh(x))
(function)
C++ documentation for atanh
-

- © cppreference.com
Licensed under the Creative Commons Attribution-ShareAlike Unported License v3.0.
- https://en.cppreference.com/w/c/numeric/complex/catanh -

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