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casinhf, casinh, casinhl

Defined in header <complex.h> 
float complex       casinhf( float complex z );
-		 (1) (since C99) 
double complex      casinh( double complex z );
-		 (2) (since C99) 
long double complex casinhl( long double complex z );
-		 (3) (since C99)
Defined in header <tgmath.h> 
#define asinh( z )
-		 (4) (since C99)
-1-3) Computes the complex arc hyperbolic sine of z with branch cuts outside the interval [−i; +i] along the imaginary axis.
-4) Type-generic macro: If z has type long double complex, casinhl is called. if z has type double complex, casinh is called, if z has type float complex, casinhf is called. If z is real or integer, then the macro invokes the corresponding real function (asinhf, asinh, asinhl). If z is imaginary, then the macro invokes the corresponding real version of the function asin, implementing the formula asinh(iy) = i asin(y), and the return type is imaginary.

Parameters

- -
z - complex argument

Return value

If no errors occur, the complex arc hyperbolic sine of z 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.

-

Error handling and special values

Errors are reported consistent with math_errhandling

-

If the implementation supports IEEE floating-point arithmetic,

-

Notes

Although 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 is asinh z = ln(z + √1+z2) For any z, asinh(z) =

-asin(iz)/i

Example

#include <stdio.h>
-#include <complex.h>
- 
-int main(void)
-{
-    double complex z = casinh(0+2*I);
-    printf("casinh(+0+2i) = %f%+fi\n", creal(z), cimag(z));
- 
-    double complex z2 = casinh(-conj(2*I)); // or casinh(CMPLX(-0.0, 2)) in C11
-    printf("casinh(-0+2i) (the other side of the cut) = %f%+fi\n", creal(z2), cimag(z2));
- 
-    // for any z, asinh(z) = asin(iz)/i
-    double complex z3 = casinh(1+2*I);
-    printf("casinh(1+2i) = %f%+fi\n", creal(z3), cimag(z3));
-    double complex 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+1.570796i
-casinh(-0+2i) (the other side of the cut) = -1.316958+1.570796i
-casinh(1+2i) = 1.469352+1.063440i
-casin(i * (1+2i))/i =  1.469352+1.063440i

References

See also

- - - - -
-
(C99)(C99)(C99)
 computes the complex arc hyperbolic cosine
(function)
-
(C99)(C99)(C99)
 computes the complex arc hyperbolic tangent
(function)
-
(C99)(C99)(C99)
 computes the complex hyperbolic sine
(function)
-
(C99)(C99)(C99)
 computes inverse hyperbolic sine (\({\small\operatorname{arsinh}{x} }\)arsinh(x))
(function)
C++ documentation for asinh
-

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Licensed under the Creative Commons Attribution-ShareAlike Unported License v3.0.
- https://en.cppreference.com/w/c/numeric/complex/casinh -

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