# Current mathematical problems of mechanics and their by A. A. Barmin, L. I. Sedov, International Conference Modern By A. A. Barmin, L. I. Sedov, International Conference Modern Mathematical Problems of Mechanics an

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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.