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# The exponents differ by an integer so for a second

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Unformatted text preview: ficients equal to zero, we find that , and , That is, for . , . Therefore one solution of the Laguerre equation is Note that if , a positive integer, then solution is a polynomial for . In that case, the ________________________________________________________________________ page 230 —————————————————————————— CHAPTER 5. —— Section 5.7 2. only for It follows that . Furthermore, and lim lim and therefore is a regular singular point. The indicial equation is given by , that is, , with roots 4. The coefficients no singular points. , 5. only for follows that and , and . are analytic for all . Furthermore, . Hence there are and It lim lim and therefore is a regular singular point. The indicial equation is given by , that is, 6. , with roots for and . and . We note that For the singularity at , , and lim lim and therefore is a regular singular point. The indicial equation is given by , that is, , with roots and . For the singularity at , ________________________________________________________________________ page 231 —————————————————————————— CHAPTER 5. —— lim lim lim and therefore lim is a regular singular point. The indicial equation is given by , that is, , with roots 7. only for follows that and . . Furthermore, and It lim lim and therefore is a regular singular point. The indicial equation is given by , that is, , with complex conjugate roots 8. Note that only for . It follows that . We find that lim , and lim lim and therefore . lim is a regular singular point. The indicial equation is given by , that is, 10. and , with roots for and lim which is undefined. Therefore at , and . We note that For the singularity at lim . , , , is an irregular singular point. For the singularity ________________________________________________________________________ page 232 —————————————————————————— CHAPTER 5. —— lim lim lim and therefore lim is a regular singular point. The indicial equation is given by , that is, 11. , with roots for and . and . We note that For the singularity at , lim lim lim and therefore , and lim is a regular singular point. The indicial equation is given by , that is, , with roots and lim . For the singularity at lim lim and therefore , lim is a regular singular point. The indicial equation is given by , that is, 12. , with roots for and . and . We note that For the singularity at , lim and therefore lim lim , and lim is a regular singular point. The indicial equation is given by ________________________________________________________________________ page 233 —————————————————————————— CHAPTER 5. —— , that is, , with roots and lim , lim lim and therefore . For the singularity at lim is a regular singular point. The indicial equation is given by , that is, 13 , with roots . Note the and and It follows that . . Furthermore, and lim lim and therefore...
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