From 633b1b05b6a89e8533a2066b88e41bee6f27cea3 Mon Sep 17 00:00:00 2001 From: Robert Date: Mon, 29 Mar 2021 17:24:44 +0200 Subject: [PATCH] Added function seqs --- assets/conv_rad.pdf | Bin 0 -> 13050 bytes chapters/multivar_calc.tex | 1 + chapters/sections/functionseq_and_diff.tex | 252 +++++++++++++++++++++ script.pdf | Bin 861864 -> 898131 bytes script.tex | 9 +- 5 files changed, 261 insertions(+), 1 deletion(-) create mode 100644 assets/conv_rad.pdf create mode 100644 chapters/sections/functionseq_and_diff.tex diff --git a/assets/conv_rad.pdf b/assets/conv_rad.pdf new file mode 100644 index 0000000000000000000000000000000000000000..b32d4f49c2e6650f192f13beeb2ff5caeba3cdaf GIT binary patch literal 13050 zcma)j1ymf%)-D=cgX;u$W^i{W1a}?W9TGISyE_Ee;O-U@+$DIBAi*_Q-jJO0-}B$| z)_Sj*>Y6QGyK2{N7JOefwUUGsGl+!~nY!`l<2^D500?j}wni2Z0ITbV! zRy3et!Vv%>Wsg}t8#}&KG;(D&uRC4Gn0QsBBG~35*xOU1zdcl*Kz9%)P;`w!`^($o zr=rvGnUC&M-7Uh?&%<1D;m0q6PY>OZ^}^rQyPs~hEEra{)}JI%1yCQ@g?(|4$hX%G zQ-0k}gnG3Ny=Pt%e7ZVI4J~PW(#$!TVK;4EA@d9_=_32}bi3sYQaG$HFEj7~>Z@|k zRbUhgwb>?e)9bJr6m|#nMxidUb>3q4v_|aOD3bMi!8!~n!-_6=Q9B<0vZym z!i2QXQle8;zZR=7fdzSct-deY8#{u-9oj87i4uKZp!=$MK{C=w^wp7hWuKQxb}C`T z>HMNV@h6%N?)gTb3GN#4Awy;@vO_ep@9bjG%_pLCh@-Aaqz5P0W@qg&4ZFr>Rqc`L zZ;8SWa`l>4?O>GpJDbY9q9)HrU)gyp$@vNKwEP*x-k&qMFh=*kV8^;q@ZaNOKL|E& zoJ2gXH-B}|C*3@N8ms2sJ0!6nV(#o(XBofhWEy_g`w3mM%pi;HC{34n3-v>&L`AfrqJ=(k=or}2xs#h#LI!3d>hv!FQE)z2|? zJH7en-#%Z$C`rDV@tJMZZ7!9@qtHnFBUd@odyG1^RwY;25Yhl48Cr(A%DCGf{65NA z)^cc<`th{l_Jb#DoIm;Td!*Ah?GxT*n*#JE)YoYvix`i83OGs2s*Z-8h##J_jorP+ z69{I_7OS!`-H-S*rXxt3qX$l+;_GwCc1>e(OvCo6RrU|wSmc=%Vo&C@SJ9(I3C|?C zdx?ieBGG%9QjEg(X=*mxBuk&nraU7U?1xDldwM$igpO3GRnCkR>c2R$6p~ISAGCyC zs^8SQP-C+f*&xk&qh{+Zof}$N6eA}EMkzy2RsM9&!}{fs4O*q4E_M1>ZWpuY)D|N^ zF6;F|)i}GB=`YTu{Pm`tlHBeUe8=78i4ozJ_Yc=2=M(GhdTqw{Gn%(zO)aR5?3QLD zvXk7(i3cWo44^~(Mr zgElF$bj_T2)j3IP9jwCX7OEh-F9V!vUN$K@{U=R^^C$=th-R8$eTI zkK)Q`)^P?zDxWlY9tr?3grYTjQx+~o;rkX9!l(WlS;|*)m*|okVIY>f&pM4744WkN)2Ee8#RO4neVkmnGX|_2EJ_*+vI8R$cno}a zTr{FA?DIIhZ&DlhR?E}}+%FJXptZs@NsTvP4}#VuvXjck1`OI5`HTv+ zwBfQjUex$uer9VW70c7o7A6dm;T5qx#)L`F>;q6qI=Si7BKk|c%`ujdd&Bs$AF~0~ zLo6;S-%dg@j*1M~X%!0&Ie%UR(_@9bZ!-iChpUVu>h1yfTsf_58Z0G_*%j1iC5>sB zuTE<14^#UCF(KiW>FC{AwkoAL28u8yyc3cpFvT6tU+$vS9^;+j+ir@mVc!5uJWE)i z7NJ)sr_k_k+jvAhi|an|Mg#-193`H{vPJb-ox?DYY=P$0^_$*_%-5v`*>w`q4x@ zW)hy(|x` zM}lYwOPjlokGmAd)FoN9r|ieCSMPJx z20u-~VGEk!czYw|l-D-hW|-9wd&VIq9{$y1++S;KN0%*f%A=e zy{M$kO?)|0i(64taBqiRSQEggN{bC%clL$F7s@1LWpqS`a8Yvq&63if92KuMuwEU7 zl&X=x1P%gywWy-fan2(1tXt()O<}Yveg<<#zH35E@{1SoRLUT($Z_^wT=9>Wn~*zD z-*ui}#8}MWOVdU=v!xl<5%p;kT)0cP@#`-EG|q84h@Ru?0mY4yC^RMhc&^KHoVp~| zc}-n8HZnRzPHuI4=)p44D1zZ*2!qxfs?&6dkNE<+(*bhZNsMMprt3}HE-bn~hXA4X zZ5*u27Y?P{_qtP3Eg3XWXOh>h<1YiUIF#)Y*0JCetq!3u!&-}xL>4Wk#Gr$GG)Qj! 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Then according to the intermediate value theorem, + \begin{equation} + \forall n \in \natn ~\exists \xi_n \in \oline = \set[{t \in [0, 1]}]{x + t(y - x)} + \end{equation} + such that + \begin{equation} + f_n(y) - f_n(x) = Df_n(\xi_n)(y - x) + \end{equation} + We have $\xi_m \in \oball[\delta](x)$. Then + \begin{equation} + \begin{split} + &\frac{1}{\norm{y - x}} \left| f_n(y) - f_n(x) - Df_n(x)(y - x) \right| \\ + = &\frac{1}{\norm{y - x}} \underbrace{\abs{(Df_n(\xi_n) - Df_n(x))(y - x)}}_{\norm{Df_n(\xi_n)} - Df_n(x)\norm{y - x}} \\ + \le &\norm{Df_n(\xi_n) - Df_n(x)} \\ + \le &\norm{Df_n(\xi_n) - g(\xi_n)} + \norm{g(\xi_n) - g(x)} + \norm{g(x) - Df_n(x)} \\ + \le &\supnorm{Df_n - g} + \norm{g(\xi_n) - g(x)} + \supnorm{g - Df_n} \\ + = &2\supnorm{Df_n - g} + \norm{g(\xi_n) - g(x)} < \epsilon + \end{split} + \end{equation} + For $n \rightarrow \infty$ we have + \begin{equation} + \frac{1}{\norm{y - x}} \abs{f(y) - f(x) - g(x)(y - x)} < \epsilon ~~\forall y \in \oball[\delta](x) + \end{equation} + Since $\epsilon > 0$ is arbitrary, we get + \begin{equation} + \limes{y}{x} \frac{1}{\norm{y - x}} \abs{f(y) - f(x) - g(x)(y - x)} = 0 + \end{equation} + This means that $f$ is differentiable in $x$ with $Df(x) = g(x)$. +\end{proof} + +\begin{rem} + On $C_B^1(U, \realn^m)$ (the space of continuous, differentiable and bounded functions with bounded derivative) we can define a norm: + \[ + \cnorm{f} := \supnorm{f} + \supnorm{Df} + \] + Then the above theorem is equivalent to the statement that $C_B^1(U, \realn^m)$ with $\cnorm{f}$ is complete. +\end{rem} + +\begin{thm} + Let $f(x) = \sum_{n=0}^{\infty} a_n x^n$ be a power series with positive convergence radius $\rho$. + Then $f$ is differentiable on $\oball[\rho](0)$ and + \[ + f'(x) = \sum_{n=0}^{\infty} n a_n x^{n-1} + \] +\end{thm} +\begin{proof} + We need to inspect the convergence radius $R$ of + \begin{equation} + \sum_{n=0}^{\infty} n a_n x^{n-1} = \frac{1}{x} \sum_{n=0}^{\infty} n a_n x^{n} + \end{equation} + $(\sqrt[n]{n})$ converges to $1$, so $\exists \epsilon > 0$ such that for sufficiently big $n$ we have + \begin{equation} + (1 - \epsilon)\sqrt[n]{a_n} \le \sqrt{n a_n} \le (1 + \epsilon)^n \sqrt{a_n} + \end{equation} + and thus + \begin{equation} + \frac{1 - \epsilon}{\rho} = (1 - \epsilon) \cdot \limsupn \sqrt[n]{\abs{a_n}} \le \limsupn \sqrt[n]{\abs{n a_n}} = \frac{1}{R} \le \frac{1 + \epsilon}{\rho} + \end{equation} + So + \begin{equation} + \implies \frac{1 - \epsilon}{\rho} \le \frac{1}{R} \le \frac{1 + \epsilon}{\rho} + \end{equation} + Since this holds for every $\epsilon$, this implies $\rho = R$. + Now for $x \in \oball[\rho](0)$ set + \begin{equation} + g(x) := \series{k} n a_n x^{n - 1} + \end{equation} + Let $x \in \oball[\rho](0)$ be fixed and choose $a > 0$ such that $\abs{x} < a < \rho$. + This means that + \begin{align*} + f_N(x) := \sum_{n=0}^{N} a_n x^n && \text{and} && g_N(x) := \sum_{n=0}^{N} a_nx^{n-1} + \end{align*} + converge uniformly on $\oball[a](0)$ to $f$ and $g$. + Obviously, $f_N' = g_N$, so $f$ is differentiable and $f' = g$. + Since differentiabiility is a local property, the desired statement follows $\forall x \in \oball[\rho](0)$. +\end{proof} + +\begin{cor} + Let $f(x) = \sum_{n=0}^{\infty} a_n x^n$ be a power series with convergence radius $\rho > 0$. Then $f \in C^{\infty}(\oball[r](0))$, and + \[ + a_k = f^{(k)}(0) \cdot \inv(k!) + \] + Furthermore, the series representation (if it exists) is unique. +\end{cor} +\begin{proof} + The infinite Differentiability follows inductively from the previous theorem. Also inductively we have + \begin{equation} + f^{(k)}(x) = \sum_{n=0}^{\infty} n(n-1) \cdots (n - k + 1) a_nx^{n-k} + \end{equation} + Choose $x = 0$ and receive + \begin{equation} + f^{(k)}(0) = n (n-1) \cdots (n - k + 1) a_n + \end{equation} +\end{proof} + +\begin{eg}[Derivative of the exponential function] + \[ + (e^x)' = \sum_{n=0}^{\infty} \left(\frac{x^n}{n!}\right)' = \sum_{n=1}^{\infty} \frac{n x^{n-1}}{n!} = \sum_{n=1}^{\infty} \frac{x^{n-1}}{(n-1)!} = \sum_{n=0}^{\infty} \frac{x^n}{n!} = e^x + \] +\end{eg} + +\begin{rem}[Taylor Series] + We can define the Taylor series for $f: \field \rightarrow \field$ + \[ + \sum_{n=0}^{\infty} \frac{f^(n)(0)}{n!} x^n = f(x) + \] + \begin{itemize} + \item In general, this doesn't hold true for all $x$, not even for $f \in C^{\infty}$. + \item The convergence radius could be $0$ + \item There are examples of convergent Taylor series that don't converge to the initial function, e.g. + \[ + f: x \mapsto \begin{cases} + \exp\left(-\frac{1}{x}\right), & x > 0 \\ + 0, & \text{else} + \end{cases} + \] + $f$ is infinitely continuously differentiable in $0$, but the Taylor series would converge to $0$. + \end{itemize} +\end{rem} + +\begin{defi} + Let $a_{\eta} \in \field$ (Multiindex notation) be coefficients $\forall \eta \in \natn_0^d$. Then + \[ + \sum_{\eta \in \natn_0^d} a_{\eta} x^{\eta} + \] + is said to be a (formal) power series with $d$ variables. + + A function $f: U \rightarrow \field$ with $U$ neighbourhood around $0$ is said to be analytic in $0$, if and only if + \[ + \exists \epsilon > 0, a_{\eta} \in \field: ~~f(x) = \sum_{\eta \in \natn_0^d} a_{\eta} x^{\eta} ~~\forall x \in \oball(0) + \] +\end{defi} + +\begin{rem} + \begin{enumerate}[(i)] + \item The convergence of the series to $S(x)$ can be defined as follows: + $\forall \epsilon > 0 ~\exists A \subset \natn_0^d \text{ finite}$ such that $\forall B \supset A$ finite we have + \[ + \abs{\sum_{\eta \in B} a_{\eta} x^{\eta} - S(x)} < \epsilon + \] + + \item If the series converges in $(y_1, \cdots, y_n)$, then it also absolutely converges in the open cuboid + \[ + \set[\abs{x_i} < \abs{y_i} ~~\forall i \in \set{1, \cdots, d}]{x \in \realn^d} + \] + which means + \[ + \sum_{\eta \in \natn_0^d} \abs{a_{\eta}} (\abs{x_1}, \cdots, \abs{x_d})^{\eta} < \infty + \] + + \item If the power series converges on a neighbourhood $U$ around $0$, then it is infinitely differentiable and + \[ + a_{\eta} = \frac{\partial^{\eta} f(0)}{\eta!} + \] + with + \begin{align*} + \partial^{\eta} := \partial_1^{\eta_1} \partial_2^{\eta_2} \cdots \partial_d^{\eta_d} && \eta! := \eta_1! \eta_2! \cdots \eta_d! + \end{align*} + + \item The formula above is only rarely useful to calculate the Taylor series. By inverting it we can calculate the + derivative of a known series representation. E.g. + \[ + f(x) = xe^{x^2} = x \cdot \sum_{k = 0}^{\infty} \frac{(x^2)^k}{k!} = \series_{k=0}^{\infty} \frac{x^{2k + 1}}{k!} ~~\forall x \in \field + \] + $f^{(k)}(0) = 0$ is $k$ is even, and it is something else if $k$ is odd. + + \item $C^{\omega}(U)$ is the space of all analytic functions. + \[ + C(U) \supset C^1(U) \supset C^2(U) \supset \cdots \supset C^k(U) \supset \cdots \supset C^{\infty}(U) \supset C^{\omega}(U) + \] + + \item The analytic functions are closed among sums, products and concatinations. + A power series is analytic within its converges radius. + \end{enumerate} +\end{rem} + +\begin{eg} + Consider the power series + \[ + \sum_{n=0}^{\infty} (xy)^n = \sum_{\eta \in \natn_0^2} (x y)^{\eta} \cdot a_{\eta} + \] + with + \begin{align*} + &a_{\eta} = 1 \text{ if } \eta_1 = \eta_2 \\ + &a_{\eta} = 0 \text{ else} + \end{align*} + This series converges on + \[ + \set[\abs{xy} < 1]{(x, y)} + \] + to $\frac{1}{1 - xy}$. + + \begin{center} + \includegraphics[scale=0.5]{../../assets/conv_rad.pdf} + \end{center} + + So the convergence area must not necessarily be a sphere. The limit function is also defined outside of the convergence area. +\end{eg} +\end{document} \ No newline at end of file diff --git a/script.pdf b/script.pdf index 3ebb78557929cab9104372b17718b6edf4e26680..846967d82a41216146390b5d621b131c00d00598 100644 GIT binary patch delta 205704 zcmd3PcX(7q+xNR?TPh^k-83Ks1kxefcR?viZ$bb;6d_R*rHdj;Z;Es{z=$AKL8J)* zRuq%~ihwk+ph&aR#RsJ8_nVos$(gg+ygXjtcU|9KIrq%$+0EVC%)EQRb@X@7`%@BT z{Sleqsq4&2Z`d%Z_kgDc7Cjebx7+2y7^}Qlmnk2r93fXTM#*hr?~@Ois>_$7yz=Nc zqx@ccbsbhkCm9~I+nw@^gd}q3o#>dTw!H^D^Fp6=m(x4@!K~2}XSW;pRes#7mv7n< zGg|DN{>+8l_P{$GI(M!Wx7AkZ*>C!<`E_}%M<2N}{Ik6qi{c}GnR#W{uk*iM-+V)h z6~{hmdiS?~K5V_Q?+Y>0Hoaqfe0jz{W|5|nBjL7KJn|-zSsVaIkawb(db@pdyLID9`?^_Q>Sf>BOgv{dvRX9{r|a> zy?^?ic^0t5Y_}Q*oqGKcfbNk{8$$xcp zrtUw#KX7vG^{#S8&c`jL+<9cE`>&DNV`k`6HZ`1a&e&)E0{NGojTiJ-xIMSi2aR$I ztM2}7%c48$q93+D{PwazKeXIz^zQ0@@xGxo&vn?d=)>VxRxa!IgYimW>G|cJpPTJE zo~?`O)8dUSTff>=yYcG6*56#+8!*(8#(kGPYD!>#r=PNBow&N=ay2$xzk-A<@VO~`uwsz(;-)` 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