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在头文件<math.h>中定义



float complex       cacoshf( float complex z );

(1)

(since C99)

double complex      cacosh( double complex z );

(2)

(since C99)

long double complex cacoshl( long double complex z );

(3)

(since C99)

Defined in header <tgmath.h>



#define acosh( z )

(4)

(since C99)

1-3）计算一个复数值的复双弧余弦值，z其中分支沿着实轴的值小于1。

4）类型 - 通用宏：如果z有类型long double complex，cacoshl被调用。如果z有类型double complex，cacosh称为，如果z有类型float complex，cacoshf称为。如果z是真实的或整数，则宏调用相应的实函数（acoshf，acosh，acoshl）。如果z是虚构的，那么宏调用相应的复数版本并且返回类型是复杂的。

参数

z

-

复杂的论点

返回值

z区间[0; 2]中的复数双曲余弦余弦。∞）沿实轴和间隔-iπ; +iπ沿虚轴。

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，

  • cacosh(conj(z)) == conj(cacosh(z))

  • 如果z是±0+0i，结果是+0+iπ/2

  • 如果z是+x+∞i（对于任何有限的x），结果是+∞+iπ/2

  • 如果z是+x+NaNi（对于任何[1]有限x），结果是NaN+NaNi并且FE_INVALID可能会被提出。

  • 如果z是-∞+yi（对于任何有限的y），结果是+∞+iπ

  • 如果z是+∞+yi（对于任何有限的y），结果是+∞+0i

  • 如果z是-∞+∞i，结果是+∞+3iπ/4

  • 如果z是±∞+NaNi，结果是+∞+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+∞+NaNi

  • 如果z是NaN+NaNi，结果是NaN+NaNi

  • 每DR471，这只适用于非零x。如果z是0+NaNi，结果应该是NaN+iπ/2

笔记

虽然C标准把这个函数命名为“复曲线双曲余弦”，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“复反双曲余弦”，不太常见的是“复曲面双曲余弦”。

反双曲余弦是一个多值函数，需要在复平面上进行分支切分。分支切割通常放置在实轴的线段（-∞，+ 1）处。

反双曲正弦的主值的数学定义是acosh z = ln（z +√z+ 1 +√z-1）对于任何z，acosh（z）=

| √z-1 |

|:----|

| √1-z |

acos（z），或者简单地说i复数（z）在复平面的上半部分。

例

#include <stdio.h>#include <complex.h>
 int main(void){
    double complex z = cacosh(0.5);    printf("cacosh(+0.5+0i) = %f%+fi\n", creal(z), cimag(z));
 
    double complex z2 = conj(0.5); // or cacosh(CMPLX(0.5, -0.0)) in C11    printf("cacosh(+0.5-0i) (the other side of the cut) = %f%+fi\n", creal(z2), cimag(z2)); 
    // in upper half-plane, acosh(z) = i*acos(z) 
    double complex z3 = casinh(1+I);    printf("casinh(1+1i) = %f%+fi\n", creal(z3), cimag(z3));
    double complex z4 = I*casin(1+I);    printf("I*asin(1+1i) = %f%+fi\n", creal(z4), cimag(z4));}

输出：

cacosh(+0.5+0i) = 0.000000-1.047198icacosh(+0.5-0i) (the other side of the cut) = 0.500000-0.000000icasinh(1+1i) = 1.061275+0.666239i
I*asin(1+1i) = -1.061275+0.666239i

参考

  • C11标准（ISO / IEC 9899：2011）：

    • 7.3.6.1可用的功能（p：192）

    • 7.25类型通用数学<tgmath.h>（p：373-375）

    • G.6.2.1可用的功能（p：539-540）

    • G.7类型 - 通用数学<tgmath.h>（p：545）

  • C99标准（ISO / IEC 9899：1999）：

    • 7.3.6.1 cacosh功能（p：174）

    • 7.22类型通用数学<tgmath.h>（p：335-337）

    • G.6.2.1可可功能（p：474-475）

    • G.7类型 - 通用数学<tgmath.h>（p：480）

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