casinhf, casinh, casinhl
Defined in header <complex.h>
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(1) | (since C99) | |
(2) | (since C99) | |
(3) | (since C99) | |
Defined in header <tgmath.h>
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#define asinh( z ) |
(4) | (since C99) |
z
with branch cuts outside the interval [−i; +i] along the imaginary axis.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,
- casinh(conj(z)) == conj(casinh(z))
- casinh(-z) == -casinh(z)
- If
z
is+0+0i
, the result is+0+0i
- If
z
isx+∞i
(for any positive finite x), the result is+∞+π/2
- If
z
isx+NaNi
(for any finite x), the result isNaN+NaNi
and FE_INVALID may be raised - If
z
is+∞+yi
(for any positive finite y), the result is+∞+0i
- If
z
is+∞+∞i
, the result is+∞+iπ/4
- If
z
is+∞+NaNi
, the result is+∞+NaNi
- If
z
isNaN+0i
, the result isNaN+0i
- If
z
isNaN+yi
(for any finite nonzero y), the result isNaN+NaNi
and FE_INVALID may be raised - If
z
isNaN+∞i
, the result is±∞+NaNi
(the sign of the real part is unspecified) - If
z
isNaN+NaNi
, the result isNaN+NaNi
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
)
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
- C11 standard (ISO/IEC 9899:2011):
- 7.3.6.2 The casinh functions (p: 192-193)
- 7.25 Type-generic math <tgmath.h> (p: 373-375)
- G.6.2.2 The casinh functions (p: 540)
- G.7 Type-generic math <tgmath.h> (p: 545)
- C99 standard (ISO/IEC 9899:1999):
- 7.3.6.2 The casinh functions (p: 174-175)
- 7.22 Type-generic math <tgmath.h> (p: 335-337)
- G.6.2.2 The casinh functions (p: 475)
- G.7 Type-generic math <tgmath.h> (p: 480)
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 (arsinh(x)) (function) |