casinhf, casinh, casinhl

C

Language

Headers

Type support

Program utilities

Variadic function support

Error handling

Dynamic memory management

Date and time utilities

Strings library

Algorithms

Numerics

Input/output support

Localization support

Atomic operations (C11)

Thread support (C11)

Technical Specifications

Numerics

Common mathematical functions

Floating-point environment

Complex number arithmetics

Random number generation

Type-generic math

Complex number arithmetic

Types and the imaginary constant
complex _Complex_I CMPLX	imaginary _Imaginary_I I
Manipulation
cimag creal carg	cabs conj cproj
Power and exponential functions
cexp clog	cpow csqrt
Trigonometric functions
ccos csin ctan	cacos casin catan
Hyperbolic functions
ccosh csinh ctanh	cacosh casinh catanh

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 [i/2; +i/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 is x+i (for any positive finite x), the result is ++/2
• If z is x+NaNi (for any finite x), the result is NaN+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 ++i/4
• If z is ++NaNi, the result is ++NaNi
• If z is NaN+0i, the result is NaN+0i
• If z is NaN+yi (for any finite nonzero y), the result is NaN+NaNi and FE_INVALID may be raised
• If z is NaN+i, the result is +NaNi (the sign of the real part is unspecified)
• If z is NaN+NaNi, the result is NaN+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
)

For any z, asinh(z) =

asin(iz)

i

Example

Run this code

#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

cacoshcacoshfcacoshl (C99)(C99)(C99)	computes the complex arc hyperbolic cosine (function)
catanhcatanhfcatanhl (C99)(C99)(C99)	computes the complex arc hyperbolic tangent (function)
csinhcsinhfcsinhl (C99)(C99)(C99)	computes the complex hyperbolic sine (function)
asinhasinhfasinhl (C99)(C99)(C99)	computes inverse hyperbolic sine (arsinh(x)) (function)