(c) Let f-¹ be the inverse of f on I. Compute f-¹ as a power series expansion centered at 0 using the above theorem. Your answer should be a series written using sigma notation.
(c) Let f-¹ be the inverse of f on I. Compute f-¹ as a power series expansion centered at 0 using the above theorem. Your answer should be a series written using sigma notation.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![3. Finding a formula for the inverse of a function can be extremely hard even if we know the inverse exists.
Miraculously, if the function is analytic then its inverse can be explicitly computed as a power series:
Theorem. Let I be an open interval and let f be an analytic function on I.
Fix a EI and b = f(a).
1
IF Vx ≤ I, ƒ'(x) #0 THEN f-¹ exists and is also analytic on its domain. Moreover, for y near b,
where co
=
· ƒ−¹(b) = = a and
VnENT,
f-1(y)
=
∞
n=0
Cn
dn-1
Cn = lim
x+aLdcn-1
(y — b)n
-
· [(ƒ (x²) = f(a))"]] ·
You will assume¹ this theorem to compute the inverse of f(x) = xe as power series. (Try the usual
approach to finding an inverse of f. You'll quickly see it's impossible.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4f181240-c48d-4fc2-b2b4-55d6cf007c89%2Fc69b191d-eba4-45b3-b8f0-c4c4bfabce02%2Fuilvt1_processed.png&w=3840&q=75)
Transcribed Image Text:3. Finding a formula for the inverse of a function can be extremely hard even if we know the inverse exists.
Miraculously, if the function is analytic then its inverse can be explicitly computed as a power series:
Theorem. Let I be an open interval and let f be an analytic function on I.
Fix a EI and b = f(a).
1
IF Vx ≤ I, ƒ'(x) #0 THEN f-¹ exists and is also analytic on its domain. Moreover, for y near b,
where co
=
· ƒ−¹(b) = = a and
VnENT,
f-1(y)
=
∞
n=0
Cn
dn-1
Cn = lim
x+aLdcn-1
(y — b)n
-
· [(ƒ (x²) = f(a))"]] ·
You will assume¹ this theorem to compute the inverse of f(x) = xe as power series. (Try the usual
approach to finding an inverse of f. You'll quickly see it's impossible.)
Transcribed Image Text:(c) Let f-¹ be the inverse of f on I.
Compute f-¹ as a power series expansion centered at 0 using the above theorem.
Your answer should be a series written using sigma notation.
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