Eπαναληπτική άσκηση 3.

Θεωρούμε τη συνάρτηση f : \mathbb{C} \rightarrow \mathbb{C} με τις εξής ιδιότητες:

  1. f(z_{1}+z_{2})=f(z_{1})+f(z_{2}) για κάθε z_{1},z_{2} \in \mathbb{C}.
  2. f(z_{1}z_{2})=f(z_{1})f(z_{2}) για κάθε z_{1},z_{2} \in \mathbb{C}.
  3. f(x)=x για κάθε x \in \mathbb{R}.

Να αποδείξετε ότι : f(z)=z ή f(z)=\bar{z} για κάθε z \in \mathbb{C}.

One Response to Eπαναληπτική άσκηση 3.

  1. Nancy says:

    f(-y^2 )=-y^2, λόγω 3.
    ⇒f(i^2 y^2 )=i^2 y^2
    ⇒f(iy iy)=(iy)^2
    ⇒f(iy)f(iy)=(iy)^2
    ⇒[f(iy)]^2=(iy)^2
    ⇒f(iy)=±iy
    Άρα, f(z)=f(x+yi)=f(x)+f(yi)=x±iy

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