coth

Hyperbolic Cotangent Function for Numeric and Symbolic Arguments

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

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

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

A =
  -1.0373 + 0.0000i   0.0000 + 0.5774i   0.0000 - 1.7321i...
   0.0000 + 0.7975i   0.0000 - 0.0000i

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

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

symA =
[ -coth(2), (3^(1/2)*1i)/3, -3^(1/2)*1i, -coth((pi*2i)/7), 0]

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

vpa(symA)

ans =
[ -1.0373147207275480958778097647678,...
0.57735026918962576450914878050196i,...
-1.7320508075688772935274463415059i,...
0.79747338888240396141568825421443i,...
0]

Plot Hyperbolic Cotangent Function

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

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

Handle Expressions Containing Hyperbolic Cotangent Function

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

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

syms x
diff(coth(x), x)
diff(coth(x), x, x)

ans =
1 - coth(x)^2
 
ans =
2*coth(x)*(coth(x)^2 - 1)

Find the indefinite integral of the hyperbolic cotangent function:

int(coth(x), x)

ans =
log(sinh(x))

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

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

ans =
x - (pi*1i)/2 - (x - (pi*1i)/2)^3/3 + (2*(x - (pi*1i)/2)^5)/15

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

rewrite(coth(x), 'exp')

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