catanhf, catanh, catanhl - cppreference.com (original) (raw)

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

1. 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.

Contents

• 1 Parameters
• 2 Return value
• 3 Error handling and special values
• 4 Notes
• 5 Example
• 6 References
• 7 See also

[edit] Parameters

[edit] 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.

[edit] 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

[edit] 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 = .

For any z, atanh(z) =

[edit] 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+2I); printf("catanh(1+2i) = %f%+fi\n", creal(z3), cimag(z3)); double complex z4 = catan((1+2I)*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

[edit] 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)

[edit] See also