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Mapes

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I'm reading a paper on tissue cell rheology ("Viscoelasticity of the human red blood cell") that models the creep compliance of the cell (in the

[tex]J(s) = \frac{1}{As+Bs^{a+1}}[/tex]

where [itex]0\leq a\leq 1[/itex]. Since there's no closed-form inverse Laplace transform for this expression, they explore early-time ([itex]t\rightarrow 0[/itex]) and late-time ([itex]t\rightarrow \infty[/itex]) behavior by using a Taylor series expansion around [itex]s\rightarrow \infty[/itex] and [itex]s\rightarrow 0[/itex], respectively. This is said to yield

[tex]J(t)\approx \frac{t^a}{B\Gamma(a+1)}-\frac{At^{2a}}{B^2\Gamma(2a+1)}+\frac{A^2t^{3a}}{B^3\Gamma(3a+1)}[/tex]

for the early-time behavior and

[tex]J(t)\approx \frac{1}{A}-\frac{Bt^{-a}}{A^2\Gamma(1-a)}[/tex]

for the late-time behavior. However, I just can't see how these expressions arise. I know that the Laplace transform of [itex]t^a[/itex] is

[tex]L[t^a]=\frac{\Gamma(a+1)}{s^{a+1}}[/tex]

and so presumably

[tex]L\left[\frac{t^a}{\Gamma(a+1)}\right]=\frac{1}{s^{a+1}}\mathrm{,}\quad L\left[\frac{t^{-a}}{\Gamma(1-a)}\right]=\frac{1}{s^{-a+1}}[/tex]

but I can't figure out where these terms would appear in a Taylor series expansion. When I try to expand [itex]J(s)[/itex] in the manner of

[tex]f(x+\Delta x)\approx f(x) + f^\prime(x)\Delta x +\frac{1}{2}f^{\prime\prime}(x)(\Delta x)^2[/tex]

I get zero or infinity for each term. Unfortunately, Mathematica is no help in investigating an expansion around [itex]s\rightarrow\infty[/itex] or [itex]s\rightarrow 0[/itex]; it just returns the original expression. Perhaps I'm making a silly error, or perhaps the paper skipped an important enabling or simplifying step. Any thoughts?

*s*-domain) as[tex]J(s) = \frac{1}{As+Bs^{a+1}}[/tex]

where [itex]0\leq a\leq 1[/itex]. Since there's no closed-form inverse Laplace transform for this expression, they explore early-time ([itex]t\rightarrow 0[/itex]) and late-time ([itex]t\rightarrow \infty[/itex]) behavior by using a Taylor series expansion around [itex]s\rightarrow \infty[/itex] and [itex]s\rightarrow 0[/itex], respectively. This is said to yield

[tex]J(t)\approx \frac{t^a}{B\Gamma(a+1)}-\frac{At^{2a}}{B^2\Gamma(2a+1)}+\frac{A^2t^{3a}}{B^3\Gamma(3a+1)}[/tex]

for the early-time behavior and

[tex]J(t)\approx \frac{1}{A}-\frac{Bt^{-a}}{A^2\Gamma(1-a)}[/tex]

for the late-time behavior. However, I just can't see how these expressions arise. I know that the Laplace transform of [itex]t^a[/itex] is

[tex]L[t^a]=\frac{\Gamma(a+1)}{s^{a+1}}[/tex]

and so presumably

[tex]L\left[\frac{t^a}{\Gamma(a+1)}\right]=\frac{1}{s^{a+1}}\mathrm{,}\quad L\left[\frac{t^{-a}}{\Gamma(1-a)}\right]=\frac{1}{s^{-a+1}}[/tex]

but I can't figure out where these terms would appear in a Taylor series expansion. When I try to expand [itex]J(s)[/itex] in the manner of

[tex]f(x+\Delta x)\approx f(x) + f^\prime(x)\Delta x +\frac{1}{2}f^{\prime\prime}(x)(\Delta x)^2[/tex]

I get zero or infinity for each term. Unfortunately, Mathematica is no help in investigating an expansion around [itex]s\rightarrow\infty[/itex] or [itex]s\rightarrow 0[/itex]; it just returns the original expression. Perhaps I'm making a silly error, or perhaps the paper skipped an important enabling or simplifying step. Any thoughts?

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