Let
where m is a non-zero , be an example (cf. (http://planetmath.org/Cf) the cyclometric functions). We form the derivatives
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f′(x)=m√1-x2cos(marcsinx), |
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f′′(x)=-m21-x2sin(marcsinx)+mx(1-x2)√1-x2cos(marcsinx), |
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which show that f satisfies the differential equation
Differentiating this repeatedly gives the equations
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(1-x2)f′′′-3xf′′+(m2-1)f′=0, |
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(1-x2)f(4)-5xf′′′+(m2-4)f′′=0, |
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and so on. Using the sum of odd numbers 1+3+5+⋯+(2n-1)=n2 and induction on n yields the recurrence relation
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(1-x2)f(n+2)-(2n+1)xf(n+1)+(m2-n2)f(n)=0. |
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Plugging in x=0 yields
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f(n+2)(0)=(n2-m2)f(n)(0) (n=0, 1, 2,…). |
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Since f′(0)=m, we have that
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f(2n+1)(0)=m(12-m2)(32-m2)…((2n-1)2-m2), |
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whereas all even derivatives of f vanish at x=0. (Note that f is an odd function
.) Thus, we obtain the Taylor of f:
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sin(marcsinx)=m1!x+m(12-m2)3!x3+m(12-m2)(32-m2)5!x5+⋯ |
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By the ratio test, this series converges for |x|<1.