 The continued fraction converges to a meromorphic function on this domain, and it converges uniformly on any closed and bounded subset of this domain that does not contain any poles.We have

so

This particular expansion is known as Lambert's continued fraction and dates back to 1768.It easily follows that

The expansion of tanh can be used to prove that "e""n" is irrational for every integer "n" (which is alas not enough to prove that "e" is transcendental).