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In Sec. 2.4 of Inside Interesting Integrals (2015), Paul J Nahin says of $$I:=\int_0^{\pi/2}\ln (a\sin x)dx=\int_0^{\pi/2}\ln (a\cos x)dx$$that:

For many years it was commonly claimed in textbooks that these are quite difficult integrals to do, best tackled with the powerful techniques of contour integration. As you’ll see with the following analysis, however, that is simply not the case.

The real method is$$2I=\int_0^{\pi/2}\ln(\frac{a^2}{2}\sin 2x)dx=\frac{\pi}{2}\ln\frac{a}{2}+\frac{1}{2}\int_0^\pi\ln(a\sin y) dy=\frac{\pi}{2}\ln\frac{a}{2}+I,$$so $I=\frac{\pi}{2}\ln\frac{a}{2}$. Presumably Euler's calculation, which Nahin says he did for $a=1$ in 1769, didn't use the above method, or else textbooks would never have claimed the need for contour integration. So when did the above proof first surface? And when did textbooks first claim the need for complex methods?

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To the title question: In E393 (1770, p. 167) and again E499 (1780, §§6, 7, 10) — reprinted in vol. 4 of his Integral Calculus (1794) — Euler integrated the Fourier series $$ \cot\varphi =2\bigl(\sin2\varphi+\sin4\varphi+\sin6\varphi+\sin8\varphi+\text{etc.}\bigr) $$ term by term twice to obtain $$ \log(\sin\varphi)=-\log2 - \cos2\varphi - \tfrac12\cos4\varphi - \tfrac13\cos6\varphi - \tfrac14\cos8\varphi - \text{etc.} $$ and \begin{align} \int_0^{\pi/2}\log(\sin\varphi)\,d\varphi &= \Bigl[-\varphi\log2 - \tfrac12\sin2\varphi - \tfrac18\sin4\varphi - \tfrac1{18}\sin6\varphi - \tfrac1{32}\sin8\varphi-\text{etc.}\Bigr]_0^{\pi/2}\\ &=\tfrac\pi2\log\tfrac12. \end{align} Koyama and Kurokawa (2005) call this “not tricky contrary to the usual explanation.” E393 has a second proof that I haven’t deciphered. Later proofs by real methods:

  • Poisson (1823, p. 489): derive $\int_0^{\pi/2}\cos^a x\cos ax\,dx=\frac{\pi}{2^{a+1}}$ with respect to $a$ at $a=0$.
  • Ellis (1841): recognize the log of $\prod_{k=1}^n\sin^2\bigl(\frac{k}{n}\frac{\pi}2\bigr)=\frac{n}{4^{n-1}}$ in a Riemann sum for the integral.
  • Goodwin (1843): your proof, also found in Grunert (1844), Todhunter (1857), Bierens de Haan (1858, 1862), etc.

Nahin’s assertion that any book “claimed the need for complex methods” sounds like a straw man. E.g. I see no such claim near the contour integration proofs of Cauchy (1825, 1826) (he attributes the formula to Euler), Lindelöf (1905), Ahlfors (1979),...

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  • $\begingroup$ ... and did Euler use notation "$\ln$"? $\endgroup$ – Gerald Edgar Aug 17 '18 at 12:43
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    $\begingroup$ You snob :-) Euler used $l$, I have now switched to $\log$. $\endgroup$ – Francois Ziegler Aug 17 '18 at 13:46
  • $\begingroup$ @FrancoisZiegler What was the "usual explanation" Koyama and Kurokawa contradicted? Were they in agreement with Nahin here? $\endgroup$ – J.G. Aug 21 '18 at 11:40
  • $\begingroup$ @J.G. I guess so — I included the paper (open access) as the closest I’ve seen to agreement. But note “contrary to” just means “unlike”, and they still don’t quote any such claim. To treat this example in a complex analysis course is a far cry from claiming it’s “best tackled” this way, and it’s hard to imagine what could motivate any mathematician to say such a thing. $\endgroup$ – Francois Ziegler Aug 21 '18 at 12:22
  • $\begingroup$ @FrancoisZiegler Fair enough. I think the trouble with complex methods is courses are more interested in showing what it looks like, how you have to be careful with the contour etc. than with finding a problem that genuinely "needs" complex methods (or would otherwise be much harder to do). $\endgroup$ – J.G. Aug 21 '18 at 12:29

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