catanhf, catanh, catanhl
| Defined in header <complex.h>
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| (1) | (since C99) | |
| (2) | (since C99) | |
| (3) | (since C99) | |
| Defined in header <tgmath.h>
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| #define atanh( z ) |
(4) | (since C99) |
z with branch cuts outside the interval [−1; +1] along the real axis.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,
- catanh(conj(z)) == conj(catanh(z))
- catanh(-z) == -catanh(z)
- If
zis+0+0i, the result is+0+0i - If
zis+0+NaNi, the result is+0+NaNi - If
zis+1+0i, the result is+∞+0iand FE_DIVBYZERO is raised - If
zisx+∞i(for any finite positive x), the result is+0+iπ/2 - If
zisx+NaNi(for any finite nonzero x), the result isNaN+NaNiand FE_INVALID may be raised - If
zis+∞+yi(for any finite positive y), the result is+0+iπ/2 - If
zis+∞+∞i, the result is+0+iπ/2 - If
zis+∞+NaNi, the result is+0+NaNi - If
zisNaN+yi(for any finite y), the result isNaN+NaNiand FE_INVALID may be raised - If
zisNaN+∞i, the result is±0+iπ/2(the sign of the real part is unspecified) - If
zisNaN+NaNi, the result isNaN+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 |
| 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
- 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
| (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 (artanh(x)) (function) |
| C++ documentation for atanh
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