By A. A. Barmin, L. I. Sedov, International Conference Modern Mathematical Problems of Mechanics an
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Additional info for Current mathematical problems of mechanics and their applications
The ﬁgure shows the graph of F3 (z) for real z > 0. 1) is called stable if all zeros lie in the left half plane Re(r) < 0. The concept is important in the theory of linear diﬀerential equations with constant coeﬃcients, as illustrated in the example below. Example 23. To solve the diﬀerential equation y¨ + 3y˙ + 2y = 0 where y˙ denotes the derivative with respect to the real variable t, one solves the corresponding characteristic equation Q2 (r) := r2 + 3r + 2 = 0. It has the roots r1 = −1 and r2 = −2.
The hypergeometric function 2 F1 (a, b; c; z) is the analytic function with power series expansion ∞ (a)n (b)n z n ab z a(a + 1)b(b + 1) z 2 =1+ + + ··· (c)n n! c 1! c(c + 1) 2! 2) at 0. Many useful special functions are special cases of 2 F1 (a, b; c; z). 2) is an inﬁnite power series whose radius of convergence is 1. 2) by comparing the power series on both sides term by term: 2 F1 (a, b; c; z) = 2 F1 (a, b + 1; c + 1; z) a(c − b) z 2 F1 (a + 1, b + 1; c + 2; z) c(c + 1) 2 F1 (a, b + 1; c + 1; z) = 2 F1 (a + 1, b + 1; c + 2; z) − − (b + 1)(c − a + 1) z 2 F1 (a + 1, b + 2; c + 3; z).
1) For simplicity we assume that z = −1. Then its classical approximants fn (z) can be written f1 (z) = f2 (z) = f3 (z) = z(1 + z) z = , 1−z 1 − z2 z z(1 − z) z z(1 − z 2 ) = = , 1−z +1−z 1 − z + z2 1 + z3 z z z z(1 + z 3 ) , = 1−z +1−z +1−z 1 − z4 and by Problem 13 on page 49 with x = −z and y = 1, fn (z) = z(1 − (−z)n ) z + (−z)n+1 = . 1 − (−z)n+1 1 − (−z)n+1 We therefore distinguish between two cases: 0 < |z| < 1 : The continued fraction converges to z. Since fn (z) = z + (−z)n+1 + higher powers of z it corresponds at 0 to the series z + 0z 2 + 0z 3 + · · · |z| > 1 : The continued fraction converges to −1.