Lindemann showed that pi is transcendental by showing that if the complex number z is algebraic (the root of an integral polynomial) then ez +1 cannot be zero.
Lindemann also claimed, and Weierstrass proved rigorously, that this result can be generalised to the following: if each of z1, … , zn is algebraic, and N1, … , Nn are algebraic and non-zero, then
cannot be zero.
It follows immediately that if z is algebraic and non-zero, ez, sin z and cos z are transcendental and corresponding results can be obtained for the inverse functions.
Although proofs of the transcendence of pi are found in many texts, for example the proof in Alan Baker’s book ‘Transcendental Number Theory’ discussed in the previous post, full proofs of the Lindemann-Weierstrass Theorem are harder to find. Baker provides a two-page proof, which is widely cited, but it is extremely condensed and key steps are omitted. I have been forced to go back to Weierstrass’ original 1885 paper to find a full proof.
Weierstrass’ paper was published in the rather fulsomely titled ‘Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin’, pages 1067 to 1085. A link to the journal is provided on the ‘Journals’ page.
Weierstrass’ paper is in three parts, and in the next three posts I will discuss each of these parts in turn.