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
\begin{equation} \frac a2\log\left(\frac{q+1}{q-1}\right)=p." \end{equation}




This is the part I feel icky about.






My current interpretation of this is that when we choose the initial condition q=\infty and p=0, we get C=\frac a2\log(-1), so that



\begin{align} \frac{a}{2} \log\left(\frac{1+q}{1-q}\right)&=p+\frac a2\log(-1)\\ \frac{a}{2} \left(\log\left(\frac{1+q}{1-q}\right)+\log(-1)\right)&=p\\ \frac{a}{2} \log\left(\frac{1+q}{1-q}(-1)\right)&=p\\ \frac{a}{2} \log\left(\frac{q+1}{q-1}\right)&=p. \end{align}
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 \mathbb{C} then we can't just freely use the sum-to-product rule
\begin{equation} \log(zw)=\log(z)+\log(w). \end{equation}



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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