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Greetings from Ireland
In relation to previous posting under the Subject Heading:-
Why not Tanh(3x) for Hyperbolic Cubic Equations
>From having studied the replies to my original query about how Tanh
could be used to solve cubic equations, i realise that the manner in
which I posed the question made it more difficult to reply to. Aside
from tanh never being EXACTLY equal to +-1, I can thanks to Gene Ward
Smith's reply see that I should have posed the question in terms of
Tanh(3z) rather than Tanh(3x) as x itself is the variable that I want
to solve for in x^3 +px +q =0.
I'm finding that i'm both understanding and verifying all the parts in
the replies, but at the same time i'm not seeing the overall connection
with the p & q in the x^3 +px +q =0.
When the substitutions are made for tanh(z) & tanh(3z) we end up with
an x^2 term that is not in the x^3 +px +q =0 that i'm trying to
co-relate it with.
Maybe what i'm saying is that the explanation needs to be almost
idiot-proof for me.
To help clarify matters i have posted the 185kb W.T.Short's pdf at
http://homepage.eircom.net/~perseus/hyperbolic.pdf
That 3-page pdf deals with all of the hyperbolic solutions for cubics
that do NOT involve the one that i'm interested in namely tanh. (he
uses p & q as well although he takes y instead of x as the variable)
If anyone cares to consult this they will see that p & q are set equal
to their hyperbolic counterparts in whetver hyperbolic identity is
being used for the particular cubic, be it cosh or sinh. He does NOT
seem to know about the Tanh solution for the range of values between
but NOT including + - 1.
In all other cases including cos that he uses in place of Tanh he
relates something to p and then substitues that relationship into the
inverse Sinh(3z) or inverse cosh(3z) that is being deployed.
Again thanks to Gene Ward Smith for confirming that it is actually
possible to do so.
While i have no doubt that many will say that Gene's answer is
sufficiently clear i must admit to still being a little bit short in
grasping its connection to x^3 +px +q =0.
There seem to be no web pages at all dealing with the hyperbolic
solution of a cubic using Tanh, so you can well appreciate how pleased
i was to learn from Gene
'that this form is particularly suitable for solving cubic equations
with one real root'
Just one person in the sci.math.research made a reply concerning this
mostly unknown tanh solution for a cubic equation! Perhaps there's a
message of somekind there for such as could interpret that or indeed
the message itself.
My thanks also to Vladimir from Bulgaria who has communicated
privately.
Sincerely
Padraigh O Searcaigh
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