asech

Inverse Hyperbolic Secant Function for Numeric and Symbolic Arguments

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

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

A = asech([-2, 0, 2/sqrt(3), 1/2, 1, 3])

A =
   0.0000 + 2.0944i      Inf + 0.0000i   0.0000 + 0.5236i...
   1.3170 + 0.0000i   0.0000 + 0.0000i   0.0000 + 1.2310i

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

symA = asech(sym([-2, 0, 2/sqrt(3), 1/2, 1, 3]))

symA =
[ (pi*2i)/3, Inf, (pi*1i)/6, acosh(2), 0, acosh(1/3)]

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

vpa(symA)

ans =
[ 2.0943951023931954923084289221863i,...
 Inf,...
 0.52359877559829887307710723054658i,...
 1.316957896924816708625046347308,...
 0,...
 1.230959417340774682134929178248i]

Plot Inverse Hyperbolic Secant Function

Plot the inverse hyperbolic secant function on the interval from 0 to 1.

syms x
fplot(asech(x),[0 1])
grid on

Handle Expressions Containing Inverse Hyperbolic Secant Function

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

Find the first and second derivatives of the inverse hyperbolic secant function. Simplify the second derivative by using simplify.

syms x
diff(asech(x), x)
simplify(diff(asech(x), x, x))

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

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

Find the indefinite integral of the inverse hyperbolic secant function:

int(asech(x), x)

ans =
atan(1/((1/x - 1)^(1/2)*(1/x + 1)^(1/2))) + x*acosh(1/x)

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

taylor(asech(x), x, Inf)

ans =
(pi*1i)/2 - 1i/x - 1i/(6*x^3) - 3i/(40*x^5)

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

rewrite(asech(x), 'log')

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