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(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
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.)
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.)
(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.
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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