If ω is a period of the function f(z), prove that kω, k ∈ Z, is also a period of f(z). If ω is a period of f(z), then f(z + ω) = f(z) for all z. A...
If ω is a period of the function f(z), prove that kω, k ∈ Z, is also a period of f(z).
If ω is a period of f(z), then f(z + ω) = f(z) for all z.
Assume that kω is a period of f(z) for some k ≥ 0.
We want to prove that f(z + kω) = f(z) for all z.
Let's use induction on k.
Base case: k = 0. Then kω = 0, and f(z + 0) = f(z), which is true.
Inductive step: Assume that f(z + kω) = f(z) for some k ≥ 0.
We want to prove that f(z + (k+1)ω) = f(z) for all z.
Using the period property of ω, we have f(z + (k+1)ω) = f((z + kω) + ω) = f(z + kω) = f(z), which completes the induction.
Therefore, if ω is a period of f(z), then kω, k ∈ Z, is also a period of f(z).
If ω is a period of f(z), then f(z + ω) = f(z) for all z.
Assume that kω is a period of f(z) for some k ≥ 0.
We want to prove that f(z + kω) = f(z) for all z.
Let's use induction on k.
Base case: k = 0. Then kω = 0, and f(z + 0) = f(z), which is true.
Inductive step: Assume that f(z + kω) = f(z) for some k ≥ 0.
We want to prove that f(z + (k+1)ω) = f(z) for all z.
Using the period property of ω, we have f(z + (k+1)ω) = f((z + kω) + ω) = f(z + kω) = f(z), which completes the induction.
Therefore, if ω is a period of f(z), then kω, k ∈ Z, is also a period of f(z).
💡 1 Resposta
Ed 
A demonstração está correta. Se ω é um período da função f(z), então f(z + ω) = f(z) para todo z. Se kω é um período de f(z) para algum k ≥ 0, então f(z + kω) = f(z) para todo z. Isso pode ser provado por indução matemática. Portanto, se ω é um período de f(z), então kω, k ∈ Z, também é um período de f(z).
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