| 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 asinh( z ) | (4) | (since C99) |
z with branch cuts outside the interval[−i; +i] along the imaginary axis.z has typelongdoublecomplex,casinhl is called. ifz has typedoublecomplex,casinh is called, ifz has typefloatcomplex,casinhf is called. Ifz is real or integer, then the macro invokes the corresponding real function (asinhf,asinh,asinhl). Ifz is imaginary, then the macro invokes the corresponding real version of the functionasin, implementing the formulaasinh(iy) = i asin(y), and the return type is imaginary.Contents |
| z | - | complex argument |
If no errors occur, the complex arc hyperbolic sine ofz is returned, in the range of a strip mathematically unbounded along the real axis and in the interval[−iπ/2; +iπ/2] 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+0iz isx+∞i (for any positive finite x), the result is+∞+π/2z isx+NaNi (for any finite x), the result isNaN+NaNi andFE_INVALID may be raisedz is+∞+yi (for any positive finite y), the result is+∞+0iz is+∞+∞i, the result is+∞+iπ/4z is+∞+NaNi, the result is+∞+NaNiz isNaN+0i, the result isNaN+0iz isNaN+yi (for any finite nonzero y), the result isNaN+NaNi andFE_INVALID may be raisedz isNaN+∞i, the result is±∞+NaNi (the sign of the real part is unspecified)z isNaN+NaNi, the result isNaN+NaNiAlthough the C standard names this function "complex arc hyperbolic sine", 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 sine", and, less common, "complex area hyperbolic sine".
Inverse hyperbolic sine is a multivalued function and requires a branch cut on the complex plane. The branch cut is conventionally placed at the line segments(-i∞,-i) and(i,i∞) of the imaginary axis.
The mathematical definition of the principal value of the inverse hyperbolic sine isasinh z = ln(z +√1+z2
)
| asin(iz) |
| i |
#include <stdio.h>#include <complex.h> int main(void){doublecomplex z= casinh(0+2*I);printf("casinh(+0+2i) = %f%+fi\n",creal(z),cimag(z)); doublecomplex z2= casinh(-conj(2*I));// or casinh(CMPLX(-0.0, 2)) in C11printf("casinh(-0+2i) (the other side of the cut) = %f%+fi\n",creal(z2),cimag(z2)); // for any z, asinh(z) = asin(iz)/idoublecomplex z3= casinh(1+2*I);printf("casinh(1+2i) = %f%+fi\n",creal(z3),cimag(z3));doublecomplex z4=casin((1+2*I)*I)/I;printf("casin(i * (1+2i))/i = %f%+fi\n",creal(z4),cimag(z4));}
Output:
casinh(+0+2i) = 1.316958+1.570796icasinh(-0+2i) (the other side of the cut) = -1.316958+1.570796icasinh(1+2i) = 1.469352+1.063440icasin(i * (1+2i))/i = 1.469352+1.063440i
(C99)(C99)(C99) | computes the complex arc hyperbolic cosine (function)[edit] |
(C99)(C99)(C99) | computes the complex arc hyperbolic tangent (function)[edit] |
(C99)(C99)(C99) | computes the complex hyperbolic sine (function)[edit] |
(C99)(C99)(C99) | computes inverse hyperbolic sine (\({\small\operatorname{arsinh}{x} }\)arsinh(x)) (function)[edit] |
C++ documentation forasinh | |
