# How to solve trascendental equation with a singularity [closed]

I have to solve a trascendental equation that behaves as

-csch(ax) == x-3


which grafically looks like

As you can see, when $a$ gets bigger the csch gets closer to the axis. Since it is divergent for $x=0$, I can't plot it at $x=0$ exactly so eventually, as $a$ gets bigger and bigger I will miss the lower intersection. The red line is a plot for $a=1000000$ and, as you can see, I can't really see the lower intersection anymore, altough I know that it has to be there somewhere.

Note: The function I'm working its not a $csch(x)$ but a complicated integral that has similar behaviour. My problem is that I don't actually know that the lower intersection exists for all values of $a$. Opposite to the $csch(x)$ example where I know the function analitically, I only know the integral numerically so when $a$ gets really big I don't know if the lower intersection trully dissapears or if it is there but I need to get closer to $x=0$.

How can I improve the precision by which I obtain the lower intersection up to bigger values of a?

• This looks like a question about some language other than Mathematica. Commented Apr 3, 2018 at 22:43

Prepend[Table[
{a, x /. Solve[-Csch[a*x] == x - 3, x, Reals][[1]] // N}, {a,
10^Range[0, 8]}],
Style[#, 14, Bold] & /@ {"a", "x"}] // Grid[#, Alignment -> Right] &


EDIT: Using FindRoot

Prepend[Table[
{a, x /. FindRoot[-Csch[a*x] == x - 3, {x, 10^-(1 + Log10[a])}]}, {a,
10^Range[0, 8]}],
Style[#, 14, Bold] & /@ {"a", "x"}] // Grid[#, Alignment -> Right] &


With other equations, if WorkingPrecision is a problem then specify a value rather than using machine precision (default).

• Thanks! What if x<0 is not a valid domain of the function? How would you rescrict the domain of the Solve/FindRoot ? Commented Apr 3, 2018 at 19:36
• You can include constraints in Solve or Reduce or NSolve. With FindRoot you have to use a different starting point. Commented Apr 3, 2018 at 19:43