In Euler's paper "De Fractionibus Continuis Dissertatio" (English Translation) he proves that the constant e≈2.71828 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
a1−q2dq=dpa2log(1+q1−q)=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+1q−1)=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+q1−q)=p+a2log(−1)a2(log(1+q1−q)+log(−1))=pa2log(1+q1−q(−1))=pa2log(q+1q−1)=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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