# Taylor series expansion and Laplace transform final value theorem

I cant figure out how some transformations are made in one article on physics. Here is expression in s-domain and they want to find its asymptotic value. $$\xi(s) = \nu_1(s+1)=\frac{1}{(s+1)} (1+\frac{2h(s+1)}{1-2g(s+1)}) (*)$$ Here is quote from the text: Asymptotic value may be obtained from equation $(*)$ by application of Laplace transform final value theorem. If the functions $(s+2)h(s+1)$ and $(s+2)(1-2g(s+2))$ (???) are expanded in Taylor series around s=0 retaining only first power terms. Using abbreviations: $$g_2=dg/ds|_{s=1}$$ $$h_1=2h(1)$$ $$h_2=-d(2^{2-s}h(s))/ds|_{s=1}$$ we can get $$\xi^{\infty}_1=\frac{g_1}{h_2}$$ $$\Delta \xi^{\infty}=\frac{g_2}{h_2}-(ln2 - \frac{1}{2})\xi^{\infty}_1 -1$$ final value can be written as $$\lim_{u \to \infty} \xi_1(u)= \xi^{\infty}_1-\Delta \xi^{\infty}e^{-u}$$ So, apparently the want to expand enumerator and denominator of (*) in Taylor series separetly (are they ? im terribly confused). And then apply final value theorem like: $$\lim_{s \to 0} s\xi(s)$$ I'm trying to do it and cant get anything similar to their answer. Could anyone please help me ? also it is known that $h(0)=1/2; g(1)=1/2; g(1)=1$.

• This is pretty hard to parse, for me. Could you give the name and authors of the article? – Antonio Vargas Aug 28 '14 at 15:24
• Sure. M. T. Robinson The influnce of the scattering law on the radiation damage displacement cascade. 1965 Philosophical Magazine. – user1364012 Aug 28 '14 at 16:37
• @Antonio i'm sorry for bothering you, but is there a chance for a help ? – user1364012 Aug 31 '14 at 6:55
• I wasn't able to find a version of the paper online that my university had access to. Do you know if it's freely available somewhere? – Antonio Vargas Aug 31 '14 at 13:21
• i doubt it, but i can send it to you by mail. – user1364012 Aug 31 '14 at 15:54