19.1 Sum and product, composition and inverse[2D6]
- E406
[1KQ]Prerequisites:2.Consider power series
\[ f(x)=∑_{n=0}^∞ a_ n x^ n ~ ~ ,~ ~ g(x)=∑_{m=0}^∞ b_ m x^ m~ , \]with non-zero radius of convergence, respectively \(r_ f\) and \(r_ g\).
Show that the product function \(h(x)=f(x)g(x)\) can be expressed in power series
\[ h(x)=∑_{k=0}^∞ c_ k x^ k \]where
\[ c_ k = ∑_{j=0}^ k a_ j b_{k-j}~ ; \]with radius of convergence \(r_ h≥\min \{ r_ f, r_ g\} \). (Note the similarity with Cauchy’s product, discussed in section ??)
Can it happen that \(r_ h{\gt}\min \{ r_ f, r_ g\} \)?
Hidden solution: [UNACCESSIBLE UUID ’1KR’]
- E406
[1KS]Prerequisites:1.Difficulty:*.Let \(g(z)=∑_{m=0}^∞ b_ m z^ m\) with \(b_ 0=g(0)≠ 0\). Express formally the reciprocal function \(f(x)=1/g(x)\) as a power series and calculate the coefficients starting from the coefficients \(b_ m\). If the radius of convergence of \(g\) is non-zero show that the radius of convergence of \(f\) is non-zero and that \(f(x)=1/g(x)\) where the two series \(f(x),g(x)\) converge. Hidden solution: [UNACCESSIBLE UUID ’1KT’]
- E406
[1KV]Prerequisites:2,3.Difficulty:*.
Consider the power series
\[ f(x)=∑_{n=0}^∞ a_ n x^ n ~ ~ ,~ ~ g(x)=∑_{m=0}^∞ b_ m x^ m~ , \]with non-zero radius of convergence, respectively \(r_ f\) and \(r_ g\). Suppose \(g(0)=0=b_ 0\). Let \(I_ f,I_ g⊂ ℂ\) be disks centered in zero with radii less than \(r_ f\) and \(r_ g\), respectively: the previous series therefore define functions \(f:I_ f→ℂ\) and \(g:I_ g→ℂ\). Up to shrinking \(I_ g\), we assume that \(g(I_ g)⊂ I_ f\).
Show that the composite function \(h=f◦ g:I_ g→ℂ\) can be expressed as a power series \(h(x)=∑_{k=0}^∞ c_ k x^ k \) (with radius of convergence at least \(r_ g\)). Show how coefficients \(c_ k\) can be computed from coefficients \(a_ k,b_ k\). Hidden solution: [UNACCESSIBLE UUID ’1KW’][UNACCESSIBLE UUID ’1KX’][UNACCESSIBLE UUID ’1KY’]
- E406
[1KZ] Difficulty:*.Let \(g(z)=∑_{m=0}^∞ b_ m z^ m\) with non-zero radius of convergence \(r_ g\). Let \(I_ g⊂ ℂ\) be a zero-centered disk of radius less than \(r_ g\); so we defined a function \(g:I_ g→ℂ\). We assume \(g(0)=0\) and \(g'(0)≠ 0\). Assuming that the inverse \(f(y)=g^{-1}(y)\) can be expressed in Taylor series \(f(x)=∑_{n=0}^∞ a_ n x^ n\), compute the coefficients of the series of \(f\) starting from those of \(g\).
Hidden solution: [UNACCESSIBLE UUID ’1M0’]
- E406
[1M1] Prerequisites:4.Difficulty:**.
Defining \(f(x)=∑_{n=0}^∞ a_ n x^ n\) where the coefficients \(a_ n\) were derived in the previous exercise 4, Try to show that the radius of convergence \(f\) is positive. 1