- #1
MathematicalPhysicist
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i need to prove that if f and and g are analytic functions in (-a,a) then so is fg.
well basically i need to find the radius of convergence of fg, which its coefficient is: [tex]c_n=\sum_{i=0}^{n} b_i*a_{n-i}[/tex], by using cauchy hadamard theorom for finding the radius of convergence, and to show that it's not greater than a.
well limsup |c_n|^1/n, then [tex]c_n=a_0b_n+...+a_nb_0[/tex]
now i think that [tex](c_n)^{\frac{1}{n}}<= ((n+1)(\max_{n \in N}(|a_n|,||b_n|))^2)^{1/n}[/tex]
i think this inequality also applies to alternating sequences.
anyway i don't know how bound it below.
well basically i need to find the radius of convergence of fg, which its coefficient is: [tex]c_n=\sum_{i=0}^{n} b_i*a_{n-i}[/tex], by using cauchy hadamard theorom for finding the radius of convergence, and to show that it's not greater than a.
well limsup |c_n|^1/n, then [tex]c_n=a_0b_n+...+a_nb_0[/tex]
now i think that [tex](c_n)^{\frac{1}{n}}<= ((n+1)(\max_{n \in N}(|a_n|,||b_n|))^2)^{1/n}[/tex]
i think this inequality also applies to alternating sequences.
anyway i don't know how bound it below.
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