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Analisi (Numeri complessi (Equazioni (Algebriche (\(z=\frac{-b \pm…

- - - - \(z=\frac{-b \pm \sqrt{b^2-4ac}}{2a}\)
      - Biquadratica
    - - Separazione \(\Re \ \ \ \Im\)
      - \(\alpha z^n = \beta \overline{z}^n\)
        
        Forma trigonometrica
- - - - Assolutamente convergente
        \[\sum\limits_{i=1}^{\infty} |a_n| < +\infty \Rightarrow \sum\limits_{i=1}^{\infty} a_n <+\infty\]
      - Abel-Dirichlet
        (numeri complessi)
        \[\begin{cases} a_n \searrow\\ \lim\limits_{n\to\infty} a_n = 0\\ b_n : s_n=\sum\limits_{k=1}^{n} b_k \ limitata \\ \end{cases}\Rightarrow \sum\limits_{i=1}^{\infty} (a_n b_n)<+\infty\]
        
        Leibniz
        \[\begin{cases} a_n \searrow\\ \lim\limits_{n\to\infty} a_n = 0\\ \end{cases}\Rightarrow \sum\limits_{i=1}^{\infty} (-1)^n a_n <+\infty\]
    - - Rapporto
        \[\lim\limits_{n\to+\infty} \frac{a_{n+1}}{a_n}=L \Rightarrow \begin{cases} L<1 & \sum\limits_{i=1}^{\infty} a_n<+\infty\\ L>1 & \sum\limits_{i=1}^{\infty} a_n=+\infty\\ L=1 & \sum\limits_{i=1}^{\infty} a_n=\ ?\\ \end{cases}\]
      - Condensazione
        \[\begin{cases} \sum\limits_{i=1}^{\infty} a_n\\ a_n \searrow\\ \end{cases}\Rightarrow caratt \bigg(\sum\limits_{i=1}^{\infty} a_n \bigg)= caratt \bigg( \sum\limits_{i=1}^{\infty} (2^na_{2^n})\bigg)\]
      - Raabe
        \[\lim\limits_{n\to+\infty}\left [ n\left ( \frac{a_{n}}{a_{n+1}} -1 \right) \right]=L \Rightarrow \begin{cases} L<1 & \sum\limits_{i=1}^{\infty} a_n=+\infty\\ L>1 & \sum\limits_{i=1}^{\infty} a_n<+\infty\\ L=1 & \sum\limits_{i=1}^{\infty} a_n=\ ?\\ \end{cases}\]
      - Kummer
        \[\lim\limits_{k\to+\infty} \left ( b_{k} \frac{a_{n}}{a_{n+1}} -b_{k+1} \right) =\lambda \Rightarrow \begin{cases} \lambda<0 & \sum\limits_{i=1}^{\infty} a_n=+\infty\\ \lambda>0 & \sum\limits_{i=1}^{\infty} a_n<+\infty\\ \lambda=0 & \sum\limits_{i=1}^{\infty} a_n=\ ?\\ \end{cases}\]
      - Radice
        \[\lim\limits_{n\to+\infty} \sqrt[n] {a_n}=L \Rightarrow \begin{cases} L<1 & \sum\limits_{i=1}^{\infty} a_n<+\infty\\ L>1 & \sum\limits_{i=1}^{\infty} a_n=+\infty\\ L=1 & \sum\limits_{i=1}^{\infty} a_n=\ ?\\ \end{cases}\]
      - Confronto asintotico
      - Confronto
- - - - Soluzioni costanti
      - Soluzioni generali
- - - - \(f(x)\sim g(x)\ e \ g(x)\ limitata \Rightarrow e^{f(x)}\sim e^{g(x)}\)
    - - \(f(x)\sim g(x),\ f(x)\rightarrow 0^{+} \ o \ f(x)\rightarrow +\infty\Rightarrow log(f(x))\sim log(g(x))\)
      - \(f(x)\rightarrow 1 \Rightarrow log(f(x))\sim f(x)-1\)
- - - - Teo di Cantor
        f definita in un insieme chiuso e limitato,continua
      - Teo dell'asintoto
        f insieme illimitato continua con asintoto orizzontale o obliquo
      - f convergente all'estremo