csch

Hyperbolic Cosecant Function for Numeric and Symbolic Arguments

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

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

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

A =
  -0.2757 + 0.0000i   0.0000 + 1.0000i      Inf + 0.0000i...
   0.0000 - 1.1547i   0.0000 - 1.2790i   0.0000 - 1.0000i

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

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

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

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

vpa(symA)

ans =
[ -0.27572056477178320775835148216303,...
1.0i,...
Inf,...
-1.1547005383792515290182975610039i,...
-1.2790480076899326057478506072714i,...
-1.0i]

Plot Hyperbolic Cosecant Function

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

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

Handle Expressions Containing Hyperbolic Cosecant Function

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

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

syms x
diff(csch(x), x)
diff(csch(x), x, x)

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

Find the indefinite integral of the hyperbolic cosecant function:

int(csch(x), x)

ans =
log(tanh(x/2))

Find the Taylor series expansion of csch(x) around x = pi*i/2:

taylor(csch(x), x, pi*i/2)

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

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

rewrite(csch(x), 'exp')

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