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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 Spring '08
 Staff
 Taylor Series, lim, Complex number

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