Archive for April, 2010

Lambert’s continued fractions for tanh(x) and tan(x) revisited

April 20, 2010

In earlier posts (since deleted) I discussed some of the properties of continued fractions and showed how Lambert’s continued fractions for tanh x and tan x could be derived and used to prove that these functions are always irrational numbers for non-zero, rational x and consequently that the exponential number, e, and pi are irrational.

In what follows, I simplify and generalise those discussions. The following is an outline, the detail can be found in the pdf here – revised in May 2016.

Irrationality criteria and continued fractions

Suppose f(x) is some function that we can represent in the following manner:

f(x) = s_1(x)=\displaystyle\frac{a_1(x)}{b_1(x)+s_2(x)}

s_k(x)=\displaystyle\frac{a_k(x)}{b_k(x)+s_{k+1}(x)}     (k\geq2)

where the a_k, b_k are polynomials in x with integer coefficients and s_k has the property that it becomes rapidly smaller in absolute value for a given x as k becomes larger. We call this a continued fraction representation of f(x) and it is not too hard to derive from it the following expression


where p, q are defined recursively by

p_{-1}=1 p_0=0

q_{-1}=0 q_0=1

and for k\geq0



Clearly p_n and q_n are integral polynomials in x. If d(n) is the degree of whichever of p_n and q_n has the higher degree, and if x = u/v is rational with u, v being integers having greatest common factor 1, then if we multiply the above expression by v^{d(n)} then it takes the form

kf(x)-h = v^{d(n)}R_n(x)

where k, h are integers and we can make use of the following theorem:

f(x) is irrational if there exist integers h and k such that 0 < |kf(x) – h| < ε for any given 1 > ε > 0

Lambert’s continued fractions

Suppose f_1(x) and f_2(x) are functions that are can be expressed as even, absolutely convergent Maclaurin series. Then we can obtain a sequence of equations


If we take f_1(x)=\cosh x and f_2(x)=\sinh x/x
then it is not too hard to show that \tanh x/x can be expressed as a continued fraction of the form

\tanh x/x=\displaystyle\frac{1}{1+s_2(x)}



\tan x/x=\displaystyle\frac{1}{1+s_2(x)}


These are Lambert’s continued fractions. The process described above can be carried through for them to show that tanh x/x and tan x/x, and hence tanh x and tan x, are irrational for non-zero rational x.