acsch

Inverse Hyperbolic Cosecant Function for Numeric and Symbolic Arguments

Depending on its arguments, acsch returns floating-point or exact symbolic results.

Compute the inverse hyperbolic cosecant function for these numbers. Because these numbers are not symbolic objects, acsch returns floating-point results.

A = acsch([-2*i, 0, 2*i/sqrt(3), 1/2, i, 3])

A =
   0.0000 + 0.5236i      Inf + 0.0000i   0.0000 - 1.0472i...
   1.4436 + 0.0000i   0.0000 - 1.5708i   0.3275 + 0.0000i

Compute the inverse hyperbolic cosecant function for the numbers converted to symbolic objects. For many symbolic (exact) numbers, acsch returns unresolved symbolic calls.

symA = acsch(sym([-2*i, 0, 2*i/sqrt(3), 1/2, i, 3]))

symA =
[ (pi*1i)/6, Inf, -(pi*1i)/3, asinh(2), -(pi*1i)/2, asinh(1/3)]

Use vpa to approximate symbolic results with floating-point numbers:

vpa(symA)

ans =
[ 0.52359877559829887307710723054658i,...
Inf,...
-1.0471975511965977461542144610932i,...
1.4436354751788103424932767402731,...
-1.5707963267948966192313216916398i,...
0.32745015023725844332253525998826]

Plot Inverse Hyperbolic Cosecant Function

Plot the inverse hyperbolic cosecant function on the interval from -10 to 10.

syms x
fplot(acsch(x),[-10 10])
grid on

Handle Expressions Containing Inverse Hyperbolic Cosecant Function

Many functions, such as diff, int, taylor, and rewrite, can handle expressions containing acsch.

Find the first and second derivatives of the inverse hyperbolic cosecant function:

syms x
diff(acsch(x), x)
diff(acsch(x), x, x)

ans =
-1/(x^2*(1/x^2 + 1)^(1/2))
 
ans =
2/(x^3*(1/x^2 + 1)^(1/2)) - 1/(x^5*(1/x^2 + 1)^(3/2))

Find the indefinite integral of the inverse hyperbolic cosecant function:

int(acsch(x), x)

ans =
x*asinh(1/x) + asinh(x)*sign(x)

Find the Taylor series expansion of acsch(x) around x = Inf:

taylor(acsch(x), x, Inf)

ans =
1/x - 1/(6*x^3) + 3/(40*x^5)

Rewrite the inverse hyperbolic cosecant function in terms of the natural logarithm:

rewrite(acsch(x), 'log')

ans =
log((1/x^2 + 1)^(1/2) + 1/x)