Does this equation $e^{2x\pi}= \frac{1+x}{1-x}$ have any solution for $x\in(0,1)$

Question

Does this equation $$e^{2x\pi}= \frac{1+x}{1-x}$$ has any solution for $x\in(0,1)$

i have tried to differenciate the function $f(x)= e^{2x\pi}- \frac{1+x}{1-x}$ and tried to check whether it is increasing or decreasing . but this donot help. Any suggestion please.

Answer 1

Note that $\lim_{x\to1}e^{2\pi x}\frac{1-x}{1+x}=0$ and that $e^{2\pi x}\frac{1-x}{1+x}>1$ when $x=\frac12$. But then $e^{2\pi x}\frac{1-x}{1+x}=1$ for some $x\in(0,1)$, which means that, for such a number $x$, $e^{2\pi x}=\frac{1+x}{1-x}$.

Answer 2

Between $0$ and $1$, $\exp(2\pi x)$ increases from $1$ to $e^{2\pi}$ while $(1+x)/(1-x)$ increases from $1$ to $\infty$. Do the graphs meet? For small positive $x$, $\exp(2\pi x)\sim 1+2\pi x$ while $(1+x)/(1-x)\sim 1+2x$, so the graph of the former is above that of the latter near $0$. The opposite is true near $1$, so we can apply IVT.

Answer 3

For small $x$ the right side behaves like $e^{2x}$, so the intersection will be closer to the pole of the right side at $x=1$. Set $x=1-u$, then we have to solve $$ e^{2\pi(1-u)}=\frac{2-u}{u}\iff e^{2\pi}ue^{-2\pi u}=2-u $$ where one sees that there has to be a root in $[0,0.1]$ even from very rough estimates of the left side, as $e^{2\pi}=535.49..$ is rather large. From the power series expansion $$ e^{2\pi}(u-2πu^2+\frac{(2π)^2u^3}2-\frac{(2π)^3u^4}6\pm...)=2-u $$ we get the first approximation of the root at $u=\frac2{e^{2\pi}+1}=0.0037279..$ giving $x=0.996272...$ which is already close to the numerical solution $x=0.9961817...$.

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The "natural" fixed-point equation following this first approximation is $u=g(u)=\frac{2}{e^{2π(1-u)}+1}$. The first iterates starting from $u_0=0$ are

 k          u[k]                x[k]
 0  0                      1
 1  0.0037279237792500566  0.99627207622075
 2  0.0038161059301026236  0.9961838940698974
 3  0.0038182168482281576  0.9961817831517719
 4  0.0038182673939971775  0.9961817326060028
 5  0.003818268604319839   0.9961817313956801
 6  0.003818268633301124   0.9961817313666989
 7  0.003818268633995082   0.996181731366005
 8  0.003818268634011698   0.9961817313659883
 9  0.0038182686340120973  0.9961817313659879
10  0.0038182686340121072  0.9961817313659879