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New PDF release: Advanced mathematical methods for scientists and engineers

By Carl M. Bender

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Example text

When E n + ! the corresponding eigenfunction is t. y(x) = A He. 5) where A is an arbitrary constant and He. <) is the Hermite polynomial of degree n [He o (x) '" 1, He , (x) x. He, (x) = x' - 1, .. J. It will be shown in Example 9 of Sec. 8 that E = n +! 4). 4b) will be shown (see Example 4 of Sec. 5) to be equivalent to the seemingly weaker inhomogeneous constraints y(x) bounded as x - ± 00. 4a) bounded ness of y(x) as Ixi- 00 implies that y_O as lxi-CD Example 4 Eigenvalue problem having transcendental eigenvalues.

En, ... , then W. 3 ) "I ::; "2 It is a theorem that W. ,. constitute a linearly dependent set of functions. For the general proof of this 42 FUNDAMENTALS theorem see Prob. 11. We prove this theorem for the simple case of two functions in the next example. Example 2 Wrorukia" for two functions. Two functions a. and b, are linearly dependent for ". ,;;" ,;; '" if b, is a constant multiple of a,. To see whether the Wronskian provides a test of the linear dependence of two functions we compute w" Suppose first that b.

I + po(n)(a. b'+1 b,a. +1) = O. 3) shows that this equation is a first-order difference equation for w,,: w,,+1 = w"po(n). 5) j=1I0 where W" is an arbitrary constant. 5)aclUally holds for difference equations of any order (see Prob. 12). 5) we can see that if Wno is nonzero, then Wn (n ~ no) is also nonzero in regions in which po(n) '" 0 (n ~ no). ] Thus, so long as po(n) -+ 0, the difference equation preserves the linear independence of solutions. Example 4 Wronskian for a constant-coefficient equation.

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