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Extra resources for Applications of Padé approximation theory in fluid dynamics
A classical example was discussed by Wallin (1972) and by Baker (1973). Let us consider a function f(x) whose expansion in MacLaurin series, gives the following two-term approximation: / ( * ) = 1 + z2- (3) The four coefficients of P1 given by Fl _ A) + AlX ~ B0 + BlX W must satisfy Eqs. 1) that for N = 1 become ■^o — aoBv , Aj = a^Ba + a0Br , A2 = a2B0 + aiBi + a0B2 , (5) Some theoretical aspects of Pade approximants - 21 where A2 = B2 = 0 and a0 = 1, at = 0, a2 = 1. These three equations present four unknowns because there is a free parameter.
L we have plotted the continued-fraction approximation for n = 4 and n = 5 for such function. The continued-fraction representation (18) is equal to the Pade approximants Some theoretical aspects of Pade approximants - 27 P0°, P°, P}, PI... (19) when every member of this sequence exists and no two members are equal (under these conditions the sequence (19) is said to be normal). 2 Convergence theorems There are several theorems on Pade approximants: the most significant differences among them deal with the way in which M and JV go to infinity.
The continued-fraction representation (18) is equal to the Pade approximants Some theoretical aspects of Pade approximants - 27 P0°, P°, P}, PI... (19) when every member of this sequence exists and no two members are equal (under these conditions the sequence (19) is said to be normal). 2 Convergence theorems There are several theorems on Pade approximants: the most significant differences among them deal with the way in which M and JV go to infinity. de Montessus de Ballore (1902) considered the convergence of P$ with M fixed; he proved that if a function f(z), with z complex variable, is regular at all the points of a circle of radius R except for a finite number of poles, of total multiplicity M, then Pj£ converges, for N —► oo, to f(z) in the circle except at the poles of f(z).