From f66b8c79d692f38e5f38d7ec75faa650ef498f35 Mon Sep 17 00:00:00 2001 From: Gabriel Videtta Date: Thu, 27 Apr 2023 12:09:16 +0200 Subject: [PATCH] fix(geometria): continua a correggere gli appunti sulle azioni di gruppo --- .../Parte teorica/2023-04-27/main.pdf | Bin 0 -> 165852 bytes .../Parte teorica/2023-04-27/main.tex | 97 ++++++++++++++++++ .../main.pdf | Bin 434835 -> 434835 bytes .../main.tex | 0 .../main.pdf | Bin 183832 -> 187957 bytes .../main.tex | 72 ++++++++----- 6 files changed, 143 insertions(+), 26 deletions(-) create mode 100644 Analisi matematica 1/Parte teorica/2023-04-27/main.pdf create mode 100644 Analisi matematica 1/Parte teorica/2023-04-27/main.tex rename Geometria 1/Prodotto scalare e hermitiano/{2023-04-17, 19, Prodotti hermitiani e teorema spettrale => 2023-04-17, 19, 26, Prodotti hermitiani e teorema spettrale}/main.pdf (99%) rename Geometria 1/Prodotto scalare e hermitiano/{2023-04-17, 19, Prodotti hermitiani e teorema spettrale => 2023-04-17, 19, 26, 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zIL+;}=FPUbxrjtZLW(Vob6M^1YYaAu$jFOOPM8 zFCdYZF(N8bMU8(bXuP$lH#_wTsA9binnBYY`{0goMw#A%g)ZSz&Lk*Kp*kO;#Cffe*gz*`qEk?t?n zVm#6J%d2os3?nDXbWu{PD=LWLhY&$BjMG{6r__=M68noKkGQF65Sc5=>y*5L|A)i2(kAA{!DdTcnC=*u8<%IZ0rIC z(ZTd9VoXzuhKq;=9Y2B*W_1bX71j=-2)fuH3W0h=Z66Z2fc1+csS1lJyh0yiHT)ET zr)6duOck)i#;GcZbZ5M`99ThEp8}@n@&QHyEuPlsVkEKrFqUP)+;hNG)-RYR;8mtQ zo=EgmCQToLtg>-{1%;Fkho}%87pNFjSuO-p1QX*J?m5!Q z@enW!7a|}w{xA`dURQiZ{v#@3;sp~)6I 0, esistono intervalli I_1, ..., I_{n(e)} tali che U I_i contiene D(f) e somma |I_i| < e, allora f è integrabile. + %TODO: se per ogni e > 0, esiste una famiglia f di intervalli numerabile tale che U_{f} I_i = D(f) e somma |I_i| < e. + + \begin{definition} [integrale improprio semplice] + Si dice che l'integrale $\int_a^b f(x) \, dx$ con $a \in \RR$ è un + \textbf{integrale improprio semplice} in $b$ se + $f$ è definita e continua su $[a, b)$ e $b = \pm\infty$, + $f$ non è definita in $b$ o non è continua in $b$. Si + definisce in modo analogo un integrale improprio semplice se $b \in \RR$. \\ + + In modo più + generale, si dice che tale integrale è improprio semplice + se $f$ è integrabile in $[a, b']$ $\forall b' < b$, ma non su + $[a, b]$ + \end{definition} + + \begin{example}\nl + \li L'integrale $\int_0^1 \frac{1}{\sin(x)} \, dx$ è un + integrale improprio semplice dacché $\frac{1}{\sin(x)}$ è + definito in $1$, ma non in $0$, ed è continuo e definito su $(0, 1)$. \\ + + \li L'integrale $\int_0^\pi \frac{1}{\sin(x)} \, dx$, invece, + non è improprio semplice, dal momento che $\frac{1}{\sin(x)}$ non + è definito né in $0$ né in $\pi$. \\ + + \li L'integrale $\int_{-1}^1 \frac{1}{x} \, dx$ non è improprio semplice + poiché $\frac{1}{x}$ non è definito in $0$. + \end{example} + + \begin{definition} + Il valore di $\int_a^b f(x) \, dx$ è definito come $\lim_{b' \to b^-} \int_a^{b'} f(x) \, dx$, se esiste. + \end{definition} + + Vi sono dunque quattro comportamenti possibili dell'integrale improprio + semplice $\int_a^b f(x) \, dx$: + + \begin{enumerate}[(a)] + \item esiste ed è finito (ossia, \textbf{converge}), + + \item esiste ed è $+\infty$ (ossia, \textbf{diverge a} $+\infty$), + + \item esiste ed è $-\infty$ (ossia, \textbf{diverge a} $-\infty$), + + \item non esiste. + \end{enumerate} + + \begin{remark} Sia $f : [a, b) \to \RR$ continua con primitiva $F : [a, b) \to \RR$. Allora $\int_a^b f(x) \, dx = \lim_{b' \to b^-} [F(b')] - F(a)$. + \end{remark} + + \begin{example}\nl + \li $\int_1^{+\infty} \frac{1}{x^\alpha} = \system{+\infty & \se \alpha \leq 1, \\ \frac{1}{a-1} & \altrimenti.}$ \\ + + \li $\int_0^{1} \frac{1}{x^\alpha} = \system{+\infty & \se \alpha \geq 1, \\ \frac{1}{1-a} & \altrimenti.}$ \\ + + \li $\int_a^{+\infty} e^{-x} \, dx = e^{-a}$. + + \li $\int_0^{+\infty} \sin(x) \, dx$ non esiste. + \end{example} + + \begin{note} + Si impiega la notazione $\int_a^b f(x) \, dx \approx \int_c^d g(x) \, dx$ per indicare che i due integrali hanno lo stesso comportamento. + \end{note} + + \begin{remark}\nl + \li Il comportamento di $\int_a^b f(x) \, dx$, se $a \in \RR$, non dipende + dalla scelta di $a$. \\ + + \li Sia $f: [a, +\infty) \to \RR$ con limite $L \neq 0 \in \RRbar$ a $+\infty$. + Allora: + + \[ \int_a^{+\infty} f(x) \, dx = \system{+\infty & \se L > 0, \\ -\infty & \altrimenti.} \] \\ %TODO: dimostralo + + \li Se $f \geq 0$ in un intorno di $b$, allora $\int_a^b f(x) \, dx$ + esiste sempre e vale o $+\infty$ o un numero finito. %TODO: dimostralo + \end{remark} +\end{document} diff --git a/Geometria 1/Prodotto scalare e hermitiano/2023-04-17, 19, Prodotti hermitiani e teorema spettrale/main.pdf b/Geometria 1/Prodotto scalare e hermitiano/2023-04-17, 19, 26, Prodotti hermitiani e teorema spettrale/main.pdf similarity index 99% rename from Geometria 1/Prodotto scalare e hermitiano/2023-04-17, 19, Prodotti hermitiani e teorema spettrale/main.pdf rename to Geometria 1/Prodotto scalare e hermitiano/2023-04-17, 19, 26, Prodotti hermitiani e teorema spettrale/main.pdf index 55d02d943a596941dc524bcdf5b8ff86d929fd3d..c102668be9cf8feee6b0f327137ebda24b078e1b 100644 GIT binary patch delta 152 zcmbPyRBG~3sfHHD7N!>F7M2#)Eo{&F*v$gw;t1pqAODI5R* delta 152 zcmbPyRBG~3sfHHD7N!>F7M2#)Eo{&F*v*U#Oe_qIr@!rElYw&t`q^9 z?BE2M&I4V$62%yfvYhKx+sI5oxY~OpPg4Zhe9}hb8UgzVks?35(5QTJE#;$|gp6`5 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