Let z be a complex number
f(z) = -1 / (1 - z)
I understand that this can be represented by a power series in a disc D(0, 1)
But how can I represent it as a power series in a disc centered at 3 + 4i?
Everywhere I look on the internet, it only tells me how to solve this when z0 = 0... Thanks!
f(z) = -1 / (1 - z)
I understand that this can be represented by a power series in a disc D(0, 1)
But how can I represent it as a power series in a disc centered at 3 + 4i?
Everywhere I look on the internet, it only tells me how to solve this when z0 = 0... Thanks!
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f(z) = -1/(1 - z)
......= -1/(1 - (z - (3 + 4i)) - (3 + 4i)), adding 0 cunningly
......= -1/[(-2 - 4i) - (z - (3 + 4i))]
......= 1/[(2 + 4i) (1 - -(z - (3 + 4i))/(2 + 4i)]
......= [1/(2 + 4i)] * Σ(n = 0 to ∞) [-(z - (3 + 4i))/(2 + 4i)]^n, via geometric series
......= Σ(n = 0 to ∞) (-1)^n (z - (3 + 4i))^n / (2 + 4i)^(n+1).
I hope this helps!
......= -1/(1 - (z - (3 + 4i)) - (3 + 4i)), adding 0 cunningly
......= -1/[(-2 - 4i) - (z - (3 + 4i))]
......= 1/[(2 + 4i) (1 - -(z - (3 + 4i))/(2 + 4i)]
......= [1/(2 + 4i)] * Σ(n = 0 to ∞) [-(z - (3 + 4i))/(2 + 4i)]^n, via geometric series
......= Σ(n = 0 to ∞) (-1)^n (z - (3 + 4i))^n / (2 + 4i)^(n+1).
I hope this helps!