Is an irregular singular point 27 49 under the

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Unformatted text preview: series of , about is analytic at . Similarly, the Taylor , is . The function singular point. is also analytic at . Hence is a regular ________________________________________________________________________ page 207 —————————————————————————— CHAPTER 5. —— 15. 32 when . Since the three coefficients have no common factors, is a singular point. The Taylor series of , about , is . Hence the function is a rational function, with is analytic at lim Hence . On the other hand, lim is a regular singular point. 16. 31 when . Since the three coefficients have no common factors, is a singular point. We find that lim lim Although the function does not have a Taylor series about , note that . Hence is a regular singular point. Furthermore, is undefined at . Therefore the points are also singular points. First note that lim Furthermore, since has period lim , Therefore From above, . Note that the function in brackets is analytic near . It follows that the function is also analytic near . Hence all the singular points are regular. 318. 4 The singular points are located at , . Dividing the ODE by , we find that and . Evidently, is not even defined at . Hence is an irregular singular point. On the other hand, the Taylor series of , about , is ________________________________________________________________________ page 208 —————————————————————————— CHAPTER 5. —— . Noting that , It is apparent that is analytic at . Similarly, , which is also analytic at 42 20. and . Hence all other singular points are regular. is the only singular point. Dividing the ODE by It follows that lim lim lim , we have , lim Hence is a regular singular point. Let Substitution into the ODE results in . . That is, . It follows that Equating the coefficients to zero, we find that , , We conclude that all the can be obtained. 22. Based on Prob. are equal to zero. Hence , the change of variable, , and . is the only solution that , transforms the ODE into the ________________________________________________________________________ page 209 —————————————————————————— CHAPTER 5. —— form . Evidently, of lim 424. 8 is a singular point. Now does not exist, , that is, Under the transformation and . Since the value , is an irregular singular point. , the ODE becomes , that is, . Therefore is a singular point. Note that and . It follows that lim lim Hence 26. 46 lim , lim is a regular singular point. Under the transformation , the ODE becomes , that is, . Therefore is a singular point. Note that and It immediately follows that the limit lim . does not exist. Hence ________________________________________________________________________ page 210 —————————————————————————— CHAPTER 5. —— is an irregular singular point. 27. 49 Under the transformation , the ODE becomes . Therefore is a singular point. Note that and . We find that lim lim , but lim The latter limit does not exist. Hence lim . is an irregul...
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