catanhf, catanh, catanhl

C

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Technical Specifications

Numerics

Common mathematical functions

Floating-point environment

Complex number arithmetics

Random number generation

Type-generic math

Complex number arithmetic

Types and the imaginary constant
complex _Complex_I CMPLX	imaginary _Imaginary_I I
Manipulation
cimag creal carg	cabs conj cproj
Power and exponential functions
cexp clog	cpow csqrt
Trigonometric functions
ccos csin ctan	cacos casin catan
Hyperbolic functions
ccosh csinh ctanh	cacosh casinh catanh

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 [i/2; +i/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+i/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+i/2
• If z is ++i, the result is +0+i/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+i/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

Run this code

#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)

See also

casinhcasinhfcasinhl (C99)(C99)(C99)	computes the complex arc hyperbolic sine (function)
cacoshcacoshfcacoshl (C99)(C99)(C99)	computes the complex arc hyperbolic cosine (function)
ctanhctanhfctanhl (C99)(C99)(C99)	computes the complex hyperbolic tangent (function)
atanhatanhfatanhl (C99)(C99)(C99)	computes inverse hyperbolic tangent (artanh(x)) (function)