Differential Topology by Victor Guillemin
By Victor Guillemin
This article suits any direction with the be aware "Manifold" within the name. it's a graduate point e-book.
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Among them, the so-called Pad´e technique is widely applied. For a given series +∞ cn xn , n=0 the corresponding [m, n] Pad´e approximant is expressed by m k=0 n k=0 am,k xk , bm,k xk where am,k , bm,k are determined by the coeﬃcients cj (j = 0, 1, 2, 3, · · · , m+n). In many cases the traditional Pad´e technique can greatly increase the convergence region and rate of a given series. 12), we have the [1, 1], [2, 2] and [3, 3] Pad´e approximants t, 3t t(15 + t2 ) , , 3 + t2 15 + 6t2 respectively.
For example, consider the limit lim (x,y)→(0,0) x2 + y 2 . |x| There are a lot of approaching paths to (0, 0). For simplicity, let us consider the path y = βx, where β is a real number. It obviously holds that lim (x,y)→(0,0) x2 + y 2 = |x| 1 + β2. The limit is therefore dependent upon the approaching path to the point (0,0). 57) is dependent upon the auxiliary parameter , because the function µm,n ( ) 0 deﬁnes diﬀerent approaching paths by diﬀerent values of . According to above explanation, the function µm,n α ( ) can be used to deﬁne diﬀerent approaching paths for a limit process by diﬀerent values of α and .
And m−n µm,n α ( < 0), n−α α−n j (−1)j ) = (− ) j=0 (1 + )j . 91) t. , Proof: Write x = 1 + + −1 < t < 2 − 1, | | −2 < < 0. By the traditional Newton binomial theorem , it holds when |x| < 1 and |1 + | < 1 that (1 + t)α = (− )−α (1 − x)α = (− )−α +∞ α n (−1)n n=0 +∞ = (− )−α (−1)n n=0 α n = (−1)n lim (− ) m→+∞ t)n (1 + + m −α n=0 α n xn t)n . (1 + + The sum of the ﬁrst m terms of above series is given by m (− )−α (−1)n α n (−1)n α n n=0 m = (− )−α n=0 m = (− )−α m tj j=0 (−1)n n=j tj j=0 i=0 m m−j = (− )−α tj j=0 © 2004 CRC Press LLC n j=0 i=0 n j α n m−j m = (− )−α (1 + + t)n (1 + )n−j n j (−1)i+j α i+j (−1)i+j α j j tj (1 + )n−j j i+j j α−j i (1 + )i (1 + )i j j m α j = j=0 m m−j tj i=0 µm,n α ( ) = α−j i (−1)i n=0 α n (1 + )i (− )j−α tn , where m−n n−α µm,n α ( ) = (− ) α−n j (−1)j j=0 (1 + )j .