M 427K - Definitions and Theorems 2

M 427K - Definitions and Theorems 2 - Integrating power...

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Integrating power series If f ( x ) = n a n ( x - x 0 ) n has radius R > 0 then f can be integrated on any subinterval of ( x 0 - R, x 0 + R ) by integrating under the summation sign. Taylor series If f ( x ) = n a n ( x - x 0 ) n has radius R > 0 then a n = f ( n ) ( x 0 ) n ! . x 0 is an “ordinary point” for P ( x ) y ±± + Q ( x ) y ± + R ( x ) y = 0 if Q ( x ) P ( x ) and R ( x ) P ( x ) have Taylor series expansions about x = x 0 with positive radii of convergence. Otherwise x 0 is a “singular point”. Power series solutions about ordinary points If x 0 is an ordinary point of P ( x ) y ±± + Q ( x ) y ± + R ( x ) y = 0 then every solution y ( x ) has a Taylor series expansion about x = x 0 with positive radius of convergence. x 0 is a “regular singular point” for P ( x ) y ±± + Q ( x ) y ± + R ( x ) y = 0 if it is a singular point and ( x - x 0 ) Q ( x ) P ( x ) and ( x - x 0 ) 2 R ( x ) P ( x ) have Taylor series expansions about x = x 0 with positive radii of convergence. Indicial equation
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