By Robert C. Gunning, Hugo Rossi

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2πi C f (t) dt , (t − c)n+1 n ∈ Z+ 20 2 Basic properties of the solutions ∞ M n |z − c|n |z − c|n = <∞ ∑ rcn ρn n=1 n=1 ∞ ∞ ∑ |an ||z − c|n ≤ ∑ n=1 and the series represents an analytic function inside the circle |z − c| < ρ, and the first part of the theorem is proved. 9, we obtain |an | ≤ Mn Mn = n n, n rc κ rc n ∈ Z+ where κ = inf{|1 − s|, |1 − s/2|, |1 − s/3|, . }, which is a positive number, due to the assumption that s is not a positive integer, and M = M/κ. The power series ∞ ∑ an (z − c)n n=1 is then uniformly convergent inside the circle |z − c| < ρ = rc /M = κrc /M since ∞ M n |z − c|n |z − c|n = <∞ ∑ n κ n rcn n=1 n=1 ρ ∞ ∞ ∑ |an ||z − c|n ≤ ∑ n=1 and the series represents an analytic function inside the circle |z − c| < ρ .

N, and at infinity we denote the roots by α, β . 4 on page 12 p(z) = 1 G(z) , ψ(z) q(z) = H(z) (ψ(z))2 Lazarus Fuchs (1833–1902), German mathematician. G. 1007/978-1-4419-7020-6_3, © Springer Science+Business Media, LLC 2010 29 30 3 Equations of Fuchsian type where n ψ(z) = ∏ (z − ar ), r=1 and G(z) and H(z) are analytic functions everywhere in the complex plane (entire functions). 5 on page 24, we conclude that p(z) and q(z) do not grow faster than |z|−1 and |z|−2 , respectively, as z → ∞.

2. Finally, we conclude that if the point, z = ∞, is a regular point to the equations, the displacement theorem is unaltered except that we now do not have any displacement of the roots of the indicial equation at infinity. , ∑nr=1 ρr = 0. 1. 6). 2. 7). 3. 3. 4. Let u(z) satisfy the differential equation du(z) d2 u(z) + p(z) + q(z)u(z) = 0 dz2 dz where 1 − α1 − β1 1 − α2 − β2 + p(z) = z z−1 1 α2 β2 β α 1 1 q(z) = − − −a z(z − 1) z z−1 Under what conditions on the constants αr , βr , r = 1, 2, and a is this a differential equation of Fuchsian type with regular singular points at z = 0, 1, ∞?