Unformatted text preview: d % , we have .
. That is,
. The four and fiveterm polynomial approximations are ________________________________________________________________________
page 186 â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€” CHAPTER 5. â€”â€”
. . The fourterm approximation
on the interval
. appears to be reasonably accurate within 20. Two linearly independent solutions of Airy's equation about Applying the ratio test to the terms of
lim are ,
lim Similarly, applying the ratio test to the terms of
lim
Hence both series converge absolutely for all
21. Let % , lim
.
. Then and ________________________________________________________________________
page 187 â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€” CHAPTER 5. â€”â€” Substitution into the ODE results in
.
First write We then obtain Setting the coefficients equal to zero, it follows that for . Note that the indices differ by two, so for and Hence the linearly independent solutions of the Hermite equation about are . Based on the recurrence relation
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page 188 â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€” CHAPTER 5. â€”â€” ,
the series solution will terminate as long as is a nonnegative even integer. If
,
then one or the other of the solutions in Part
will contain at most
terms. In
particular, we obtain the polynomial solutions corresponding to . Observe that if , and , then and for . It follows that the coefficient of , in and , is for
for
Then by definition,
for
for
Therefore the first six Hermite polynomials are 23. The series solution is given by
________________________________________________________________________
page 189 â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€” CHAPTER 5. â€”â€” 24. The series solution is given by
. 25. The series solution is given by
. ________________________________________________________________________
page 190 â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€” CHAPTER 5. â€”â€” 26. The series solution is given by 27. The series solution is given by
. ________________________________________________________________________
page 191 â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€” CHAPTER 5. â€”â€” 28. Let . Then and Substitution into the ODE results in
.
After appropriately shifting the indices, it follows that
.
We find that for Since and . Writing out the individual equations, and , the remaining coefficients satisfy the equations ________________________________________________________________________
page 192 â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€” CHAPTER 5. â€”â€” That is,
,
,
of the initial value problem is , , . Hence the series solution ________________________________________________________________________
page 193 â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€...
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 Spring '08
 Staff
 Taylor Series, lim, Complex number

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