sech

Symbolic hyperbolic secant function

Syntax

sech(X)

Description

sech(X) returns the hyperbolic secant function of X.

Examples

Hyperbolic Secant Function for Numeric and Symbolic Arguments

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

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

A = sech([-2, -pi*i, pi*i/6, 0, pi*i/3, 5*pi*i/7, 1])

A =
    0.2658   -1.0000    1.1547    1.0000    2.0000   -1.6039    0.6481

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

symA = sech(sym([-2, -pi*i, pi*i/6, 0, pi*i/3, 5*pi*i/7, 1]))

symA =
[ 1/cosh(2), -1, (2*3^(1/2))/3, 1, 2, -1/cosh((pi*2i)/7), 1/cosh(1)]

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

vpa(symA)

ans =
[ 0.26580222883407969212086273981989,...
-1.0,...
1.1547005383792515290182975610039,...
1.0,...
2.0,...
-1.6038754716096765049444092780298,...
0.64805427366388539957497735322615]

Plot Hyperbolic Secant Function

Plot the hyperbolic secant function on the interval from -10 to 10.

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

Handle Expressions Containing Hyperbolic Secant Function

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

Find the first and second derivatives of the hyperbolic secant function:

syms x
diff(sech(x), x)
diff(sech(x), x, x)

ans =
-sinh(x)/cosh(x)^2
 
ans =
(2*sinh(x)^2)/cosh(x)^3 - 1/cosh(x)

Find the indefinite integral of the hyperbolic secant function:

int(sech(x), x)

ans =
2*atan(exp(x))

Find the Taylor series expansion of sech(x):

taylor(sech(x), x)

ans =
(5*x^4)/24 - x^2/2 + 1

Rewrite the hyperbolic secant function in terms of the exponential function:

rewrite(sech(x), 'exp')

ans =
1/(exp(-x)/2 + exp(x)/2)

Input Arguments

X — Input

symbolic number | symbolic variable | symbolic expression | symbolic function | symbolic vector | symbolic matrix

Input, specified as a symbolic number, variable, expression, or function, or as a vector or matrix of symbolic numbers, variables, expressions, or functions.

Version History

Introduced before R2006a