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I have

$$ f(x) = \sqrt{\ln\left(a\cosh^{2}(mx)(1+bx^{2})\right)} $$

If I expand this as a series I should get something of the form

$$ \sqrt{\ln a}+gx^{2}+\mathcal{O}(x^{4}) $$

but I'm having real difficulty getting Mathematica to give me the necessary Taylor series of $f(x)$. Can anyone help? I only really need $g$, the later coefficients aren't necessary. Perhaps there's a reason why Mathematica has a problem?

apg
  • 2,797
  • so you were unsuccessful in finding a better function than $\cosh(x)$? – Guy Mar 15 '14 at 13:19
  • For me, Mathematica immediately gave me the result using Series. I am using Mathematica 7 – Hrodelbert Mar 15 '14 at 14:04
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    A work around is to calculate $f''(0)/2$, the second-order Taylor coefficient. That also works. The answer is $g=\frac{2b+m^2}{4\ln(a)}$ – Hrodelbert Mar 15 '14 at 14:17
  • perhaps you could give me the $x^4$ term so I can work out b and m? – apg Mar 15 '14 at 19:14
  • I have a Taylor series of another equation and need to solve for the three variables b,m and a, so I actually need three equations if you can give me the third? Thanks for this – apg Mar 15 '14 at 19:48

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