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在头文件<math.h>中定义



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）计算z区间-1以外的分支切口的复数双曲正切; +1沿实轴。

4）类型 - 通用宏：如果z有类型long double complex，catanhl被调用。如果z有类型double complex，catanh称为，如果z有类型float complex，catanhf称为。如果z是真实的或整数，则宏调用相应的实函数（atanhf，atanh，atanhl）。如果z是虚构的，那么宏调用相应的真实版本atan，实现公式atanh（iy）= i atan（y），并且返回类型是虚构的。

参数

z

-

复杂的论点

返回值

如果没有出现错误，z则返回复数圆弧双曲正切值，该值在半径数值范围内沿着实轴且在区间-iπ/ 2中是数学上无界的; +iπ/ 2沿着虚轴。

错误处理和特殊值

报告的错误与math_errhandling一致。

如果实现支持IEEE浮点运算，

  • catanh(conj(z)) == conj(catanh(z))

  • catanh(-z) == -catanh(z)

  • 如果z是+0+0i，结果是+0+0i

  • 如果z是+0+NaNi，结果是+0+NaNi

  • 如果z是+1+0i，结果是+∞+0i并且FE_DIVBYZERO被提出

  • 如果z是x+∞i（对于任何有限的正x），结果是+0+iπ/2

  • 如果z是x+NaNi（对于任何有限的非零x），结果是NaN+NaNi并FE_INVALID可能被提出

  • 如果z是+∞+yi（对于任何有限的正y），结果是+0+iπ/2

  • 如果z是+∞+∞i，结果是+0+iπ/2

  • 如果z是+∞+NaNi，结果是+0+NaNi

  • 如果z是NaN+yi（对于任何有限的y），结果是NaN+NaNi并且FE_INVALID可能会被提出

  • 如果z是NaN+∞i，结果是±0+iπ/2（真实部分的符号未指定）

  • 如果z是NaN+NaNi，结果是NaN+NaNi

笔记

尽管C标准将这个函数命名为“复圆弧双曲正切”，但双曲函数的反函数是区域函数。他们的论点是双曲线领域，而不是弧线。正确的名称是“复反双曲正切”，并且不太常见的是“复曲面双曲正切”。

逆双曲正切是一种多值函数，需要在复平面上进行分支切割。分支切割通常放置在实轴的线段（-∞，-1）和[+ 1，+∞）处。反双曲正弦的主值的数学定义是atanh z =

| ln(1+z)-ln(z-1) |

|:----|

| 2 |

.

For any z, atanh(z) =

| atan(iz) |

|:----|

| i |

例

#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));}

输出：

catanh(+2+0i) = 0.549306+1.570796icatanh(+2-0i) (the other side of the cut) = 0.549306-1.570796icatanh(1+2i) = 0.173287+1.178097icatan(i * (1+2i))/i = 0.173287+1.178097i

参考

  • C11标准（ISO / IEC 9899：2011）：

    • 7.3.6.3函数（p：193）

    • 7.25类型通用数学<tgmath.h>（p：373-375）

    • G.6.2.3通信功能（p：540-541）

    • G.7类型 - 通用数学<tgmath.h>（p：545）

  • C99标准（ISO / IEC 9899：1999）：

    • 7.3.6.3 catanh函数（p：175）

    • 7.22类型通用数学<tgmath.h>（p：335-337）

    • G.6.2.3通信功能（p：475-476）

    • G.7类型 - 通用数学<tgmath.h>（p：480）

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