| Types and the imaginary constant | |||||||||||||||||||||||||||||||
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| Manipulation | |||||||||||||||||||||||||||||||
| Power and exponential functions | |||||||||||||||||||||||||||||||
| Trigonometric functions | |||||||||||||||||||||||||||||||
| Hyperbolic functions | |||||||||||||||||||||||||||||||
Defined in header <complex.h> | ||
| (1) | (since C99) | |
| (2) | (since C99) | |
| (3) | (since C99) | |
Defined in header <tgmath.h> | ||
#define acosh( z ) | (4) | (since C99) |
z with branch cut at values less than 1 along the real axis.z has typelongdoublecomplex,cacoshl is called. ifz has typedoublecomplex,cacosh is called, ifz has typefloatcomplex,cacoshf is called. Ifz is real or integer, then the macro invokes the corresponding real function (acoshf,acosh,acoshl). Ifz is imaginary, then the macro invokes the corresponding complex number version and the return type is complex.Contents |
| z | - | complex argument |
The complex arc hyperbolic cosine ofz in the interval[0; ∞) along the real axis and in the interval[−iπ; +iπ] along the imaginary axis.
Errors are reported consistent withmath_errhandling
If the implementation supports IEEE floating-point arithmetic,
z is±0+0i, the result is+0+iπ/2z is+x+∞i (for any finite x), the result is+∞+iπ/2z is+x+NaNi (for non-zero finite x), the result isNaN+NaNi andFE_INVALID may be raised.z is0+NaNi, the result isNaN±iπ/2, where the sign of the imaginary part is unspecifiedz is-∞+yi (for any positive finite y), the result is+∞+iπz is+∞+yi (for any positive finite y), the result is+∞+0iz is-∞+∞i, the result is+∞+3iπ/4z is+∞+∞i, the result is+∞+iπ/4z is±∞+NaNi, the result is+∞+NaNiz isNaN+yi (for any finite y), the result isNaN+NaNi andFE_INVALID may be raised.z isNaN+∞i, the result is+∞+NaNiz isNaN+NaNi, the result isNaN+NaNiAlthough the C standard names this function "complex arc hyperbolic cosine", 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 cosine", and, less common, "complex area hyperbolic cosine".
Inverse hyperbolic cosine is a multivalued function and requires a branch cut on the complex plane. The branch cut is conventionally placed at the line segment(-∞,+1) of the real axis.
The mathematical definition of the principal value of the inverse hyperbolic cosine isacosh z = ln(z +√z+1√z-1)
For any z,acosh(z) =| √z-1 |
| √1-z |
#include <stdio.h>#include <complex.h> int main(void){doublecomplex z= cacosh(0.5);printf("cacosh(+0.5+0i) = %f%+fi\n",creal(z),cimag(z)); doublecomplex z2=conj(0.5);// or cacosh(CMPLX(0.5, -0.0)) in C11printf("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)doublecomplex z3=casinh(1+I);printf("casinh(1+1i) = %f%+fi\n",creal(z3),cimag(z3));doublecomplex z4= I*casin(1+I);printf("I*asin(1+1i) = %f%+fi\n",creal(z4),cimag(z4));}
Output:
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+0.666239iI*asin(1+1i) = -1.061275+0.666239i
(C99)(C99)(C99) | computes the complex arc cosine (function)[edit] |
(C99)(C99)(C99) | computes the complex arc hyperbolic sine (function)[edit] |
(C99)(C99)(C99) | computes the complex arc hyperbolic tangent (function)[edit] |
(C99)(C99)(C99) | computes the complex hyperbolic cosine (function)[edit] |
(C99)(C99)(C99) | computes inverse hyperbolic cosine (\({\small\operatorname{arcosh}{x} }\)arcosh(x)) (function)[edit] |
C++ documentation foracosh | |
