Unformatted text preview: icients have no common factors,
is a singular point. Near
,
lim lim lim
Hence lim is a regular singular point. Let
. Then and Substitution into the ODE results in . That is,
.
It follows that Assuming that , we obtain the indicial equation , with roots ________________________________________________________________________
page 220 —————————————————————————— CHAPTER 5. ——
and
. It immediately follows that
equal
to zero, we have . Setting the remaining coefficients ,
For . , the recurrence relation becomes
, Since For . , the odd coefficients are zero. Furthermore, for , , the recurrence relation becomes
, Since . , the odd coefficients are zero, and for , The two linearly independent solutions are 3
5. Note that and , which are both analytic at
. Set
. Substitution into the ODE results in
, and after multiplying both sides of the equation by ,
. It follows that ________________________________________________________________________
page 221 —————————————————————————— CHAPTER 5. —— Setting the coefficients equal to zero, the indicial equation is
are
and
. Here
. The recurrence relation is
,
For . The roots . ,
, Hence for . ,
. Therefore one solution is
. 5.
3 Here and , which are both analytic at
. Set
. Substitution into the ODE results in It follows that .
Assuming
, the indicial equation is
, with roots
Setting the remaining coefficients equal to zero, we have
, and
, , . It immediately follows that the odd coefficients are equal to zero. For
,
So for , . , ________________________________________________________________________
page 222 —————————————————————————— CHAPTER 5. —— .
For ,
, So for . ,
. The two linearly independent solutions are 6. Note that and , which are both analytic at
. Set
. Substitution into the ODE results in After adjusting the indices in the secondtolast series, we obtain Assuming
, the indicial equation is
, with roots
the remaining coefficients equal to zero, the recurrence relation is
, . Setting . First note that . So for
, , . ________________________________________________________________________
page 223 —————————————————————————— CHAPTER 5. ——
It follows that
,
For . ,
, , and therefore
, . The two linearly independent solutions are 7. Here and After multiplying both sides by , which are both analytic at
. Set
. Substitution into the ODE results in , +1 +1 After adjusting the indices in the last two series, we obtain ________________________________________________________________________
page 224 —————————————————————————— CHAPTER 5. —— Assuming
, the indicial equation is
, with roots
the remaining coefficients equal to zero, the recurrence relation is
, . , With . Setting . , Hence one solution is
.
8. Note that
Set and , which are both analytic at
.
. Substitution...
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 Spring '08
 Staff
 Taylor Series, lim, Complex number

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