Unformatted text preview: into the ODE results in After adjusting the indices in the secondtolast series, we obtain Assuming
, the indicial equation is
, with roots
and
. Setting the remaining coefficients equal to zero, the recurrence relation is
, . Setting the remaining coefficients equal to zero, we have
of the odd coefficients are zero. With
,
,
So for , which implies that all
. , ________________________________________________________________________
page 225 —————————————————————————— CHAPTER 5. —— .
With ,
, So for . ,
. The two linearly independent solutions are 9. Note that
Set and , which are both analytic at
.
. Substitution into the ODE results in After adjusting the indices in the secondtolast series, we obtain Assuming
, the indicial equation is
, with roots
and
. Setting the remaining coefficients equal to zero, the recurrence relation is
,
With . , ________________________________________________________________________
page 226 —————————————————————————— CHAPTER 5. —— ,
It follows that for . ,
. Therefore one solution is
. 10. Here
Set and , which are both analytic at
.
. Substitution into the ODE results in After adjusting the indices in the second series, we obtain Assuming
, the indicial equation is
, with roots
Setting the remaining coefficients equal to zero, we find that
. The recurrence
relation is
,
With . ,
, Since , the odd coefficients are zero. So for .
,
. Therefore one solution is ________________________________________________________________________
page 227 —————————————————————————— CHAPTER 5. —— . 12 . Dividing through by the leading coefficient, the ODE can be written as
. For ,
lim lim
For lim lim ,
lim lim lim lim Hence both
and
are regular singular points. As shown in Example ,
the indicial equation is given by
.
In this case, both sets of roots are
. Let
, and
equation becomes Based on Part , and . Under this change of variable, the differential is a regular singular point. Set Substitution into the ODE results in Upon inspection, we can also write
________________________________________________________________________
page 228 —————————————————————————— CHAPTER 5. —— After adjusting the indices in the second series, it follows that Assuming that
, the indicial equation is
The recurrence relation is , with roots , . ,
With With , we find that for , we find that for . , , The two linearly independent solutions of the Chebyshev equation are 13. Here
In fact, and
lim Hence the indicial equation is , which are both analytic at
and lim
, with roots . .
. Set ________________________________________________________________________
page 229 —————————————————————————— CHAPTER 5. —— .
Substitution into the ODE results in That is, It follows that
.
Setting the coef...
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 Spring '08
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 Taylor Series, lim, Complex number

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