Friday, 6 March 2015

ordinary differential equations - What is Euler doing?

In Euler's paper "De Fractionibus Continuis Dissertatio" (English Translation) he proves that the constant e2.71828 is irrational.1 One step in the proof threw me for a loop, though. In the middle of paragraph 29, Euler solves the ODE



adq+q2dp=dp



by rearranging and integrating to get




a1q2dq=dpa2log(1+q1q)=p+C.



So far, so good. But now Euler says (quoting from Wyman's translation) "the constant ought to be determined from this equation by setting q= when p=0. Wherefore there follows
a2log(q+1q1)=p."




This is the part I feel icky about.






My current interpretation of this is that when we choose the initial condition q= and p=0, we get C=a2log(1), so that



a2log(1+q1q)=p+a2log(1)a2(log(1+q1q)+log(1))=pa2log(1+q1q(1))=pa2log(q+1q1)=p.


This interpretation leaves more than a little to be desired. For one thing, log(1) doesn't even make sense in a purely real context, but if we decide to work in C then we can't just freely use the sum-to-product rule
log(zw)=log(z)+log(w).



How should I understand this step in Euler's proof?







1) Ed Sandifer claims that this is the first rigorous proof that e is irrational (Link. Note that the linked PDF didn't display correctly in the two browsers I tried, but was fine when I downloaded it.)

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