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**Extra resources for Mathieu Functions and Spheroidal Functions and Their Mathematical Foundations: Further Studies**

**Example text**

On ~= ~. ,O). = I := Y i ( O ; Z , # 2 ) y # ( O ; ~ , y 2 ) _ (n E IN) has been shown a l r e a d y now, h o w e v e r , more g e n e r a l l y the proof of , 7(0) . Then we d e f i n e holds. 12. A ( t , y 2) with (I) with in from the considerations (~). The eigensolutions Yn(X) := y~(x) to An ...... 21. , but which are given it re- below for are (n-l)! 'pm (x) (2m+n-l)! 5) hold. 21.

We start with the following assumptions (~ Let ~ to ~ be linear spaces over ( , ) : ~ * > ( , ) : 7~7" > C : 7x7" > C ( ,> C , all of them u E ~ (u,v*) = 0 , then let C be bilinear mappings which are non-degenerate all ~{0}; v* = 0 in the following sense: if for and conversely; analogously in the other cases. ,y s are linearly independent in 7 and 7", respectively, the r - s elements spectively; for Let ~ and 7 then x ® Yo are linearly independent in 7 and 7 " , reP a E ~ , b E 7 , c* E ~ * , d* E 7 " there shall always hold (a® (•) > 7* b,c*® d*) = (a,c*) be seperated-topological <' )t : 7 × ~-* > • (b,d*) .

One s e t s for hold. 6) is the elementary is again a simple series estimate. I0. -1 and obtains < x < 1 x 1 u(x) := f YI(-~)f(g)d~ " YI(X) + f YI(~)f(~)d~ " YI(-x) -1 x and u s e s x I u,(~> = I . . d ~ . 12. Y[-I] := 0 . Then for K = 0,I,2,... 1 u(x) . for ~ ~ = 0,1,2 .... K x 2 there obviously holds the notation at 42 If now Z is precisely a k - fold zero of A(-,y 2) , then one obtains easily that either Yio] (0) Yil] (0) . . . Yik-l] (0) = O, Y[o] (0) Y[I] (0) . . . Y[k-1] (0) I y[k](O) # 0 or and conversely.