# 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?

## closed as off-topic by Daniel Lichtblau, Coolwater, José Antonio Díaz Navas, Sektor, YoungApr 8 '18 at 17:24

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question cannot be answered without additional information. Questions on problems in code must describe the specific problem and include valid code to reproduce it. Any data used for programming examples should be embedded in the question or code to generate the (fake) data must be included." – Daniel Lichtblau, Coolwater, José Antonio Díaz Navas, Sektor, Young
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• This looks like a question about some language other than Mathematica. – Daniel Lichtblau Apr 3 '18 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 ? – P. C. Spaniel Apr 3 '18 at 19:36
• You can include constraints in Solve or Reduce or NSolve. With FindRoot you have to use a different starting point. – Bob Hanlon Apr 3 '18 at 19:43