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\begin{document}

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{\bf $\stackrel{\mbox{MA330 }}{\mbox{\bf Sathaye}}$
\hfill Explanation of Landen's Paper \hfill \ \    }
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The paper by John Landen {\it (A New method of computing the sum of certain series Phil. Trans. 1759-60 (51) p. 553-565)} is being discussed in class.
Here we give an outline of the proofs.



\begin{enumerate}
\item
The notation hyp.log is equivalent to just the natural log in modern notation. 
Let $\i=\sqrt{-1}$. The paper declares values of its log and in modern notation, we may write $\log(\i)=\pi\i/2$ and $\log(-\i)=\log(1/\i) =-\log(\i)=-\pi\i/2$.

 It follows that $\log(-1)=\log(\i^2)=\pi\i$. However, $\log(-1)$, can be also viewed as $-\pi\i$ since $-1=(-\i)^2$.
The paper names $a=\pi/2$ and hence determines $\log(1/\i)= -\log(\i) =-a\i = a/\i$.
We shall simply use $\log(\i) = \pi \i/2 $ and $\log(-1) = \pm \pi\i$.

\item The main formula of interest is the series expansion
$$\log(\frac{1}{1-x})= x+x^2\2+\cdots = \sum x^n/n .$$
We shall denote it as $L_1(x)$. (It is not named in the paper.)

In general, we define $L_r(x) = \sum x^n/(n^r)$ also not named in the paper.



 

\end{enumerate}
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