Lecture slides for MA2730 Analysis I

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1 Lecture slides for MA2730 Analysis I Simon people.brunel.ac.uk/~icsrsss simon.shaw@brunel.ac.uk College of Engineering, Design and Physical Sciences bicom & Materials and Manufacturing Research Institute Brunel University November 20, 2015

2 Contents of the teaching and assessment blocks MA2730: Analysis I Analysis taming infinity Maclaurin and Taylor series. Sequences. Improper Integrals. Series. Convergence. L A TEX2ε assignment in December. Question(s) in January class test. Question(s) in end of year exam. Web Page:

3 MA L A TEX: topics for Advanced concepts: packages; macros; environments. Tables. Floats. Cross referencing. tikz graphics, The picture environnment.

4 The topics for this lecture are both wide and deep. This means that our treatment will be particularly superficial. All of the topics touched on today can be probed much further and the examples that we use made much more complicated. As ever, there are plenty of learning resources out there on the net so feel free to play around and build on what we do. We start with making our own macros... Before we start, make sure your preamble contains at least the same material as for lab 2.

5 Suppose that you wanted to write many instances of dy dx. In code this is \frac{dy}{dx} But you might still think it is a lot to type many, many times. You can get around this by defining a macro: \newcommand{\dydx}{\frac{dy}{dx}} And then: typing $\dydx$ will produce dy dx.

6 Suppose that you wanted to write many instances of dy dx. In code this is \frac{dy}{dx} But you might still think it is a lot to type many, many times. You can get around this by defining a macro: \newcommand{\dydx}{\frac{dy}{dx}} And then: typing $\dydx$ will produce dy dx.

7 Suppose that you wanted to write many instances of dy dx. In code this is \frac{dy}{dx} But you might still think it is a lot to type many, many times. You can get around this by defining a macro: \newcommand{\dydx}{\frac{dy}{dx}} And then: typing $\dydx$ will produce dy dx.

8 Suppose that you wanted to write many instances of dy dx. In code this is \frac{dy}{dx} But you might still think it is a lot to type many, many times. You can get around this by defining a macro: \newcommand{\dydx}{\frac{dy}{dx}} And then: typing $\dydx$ will produce dy dx.

9 Suppose that you wanted to write many instances of dy dx. In code this is \frac{dy}{dx} But you might still think it is a lot to type many, many times. You can get around this by defining a macro: \newcommand{\dydx}{\frac{dy}{dx}} And then: typing $\dydx$ will produce dy dx.

10 dy But what if you wanted to write several variants: dx, df dz, ds dt etc? You can abbreviate this by defining a macro with arguments: \newcommand{\deriv}[2]{\frac{d#1}{d#2}} And then: typing \[\deriv{s}{t}\deriv{t}{s}=1.\] will produce ds dt dt ds = 1. While \[\deriv{y}{x}\deriv{x}{y}=1.\] will produce dy dx dxdy = 1.

11 dy But what if you wanted to write several variants: dx, df dz, ds dt etc? You can abbreviate this by defining a macro with arguments: \newcommand{\deriv}[2]{\frac{d#1}{d#2}} And then: typing \[\deriv{s}{t}\deriv{t}{s}=1.\] will produce ds dt dt ds = 1. While \[\deriv{y}{x}\deriv{x}{y}=1.\] will produce dy dx dxdy = 1.

12 dy But what if you wanted to write several variants: dx, df dz, ds dt etc? You can abbreviate this by defining a macro with arguments: \newcommand{\deriv}[2]{\frac{d#1}{d#2}} And then: typing \[\deriv{s}{t}\deriv{t}{s}=1.\] will produce ds dt dt ds = 1. While \[\deriv{y}{x}\deriv{x}{y}=1.\] will produce dy dx dxdy = 1.

13 dy But what if you wanted to write several variants: dx, df dz, ds dt etc? You can abbreviate this by defining a macro with arguments: \newcommand{\deriv}[2]{\frac{d#1}{d#2}} And then: typing \[\deriv{s}{t}\deriv{t}{s}=1.\] will produce ds dt dt ds = 1. While \[\deriv{y}{x}\deriv{x}{y}=1.\] will produce dy dx dxdy = 1.

14 How about these? dy dx, d2 f dm+1 s dz 2, etc? dtm+1 You can deal with these by defining a macro with an optional argument: \newcommand{\derivn}[3][]{\frac{d^{#1}#2}{d#3^{#1}}} And then: typing \[\derivn{y}{x}+\derivn[2]{f}{z}+\derivn[m+1]{s}{t}=?\] will produce dy dx + d2 f dz 2 + dm+1 s =? dtm+1 You can have only one optional argument, and up to nine arguments in total. The \newcommand{...} definitions should go in the document preamble.

15 How about these? dy dx, d2 f dm+1 s dz 2, etc? dtm+1 You can deal with these by defining a macro with an optional argument: \newcommand{\derivn}[3][]{\frac{d^{#1}#2}{d#3^{#1}}} And then: typing \[\derivn{y}{x}+\derivn[2]{f}{z}+\derivn[m+1]{s}{t}=?\] will produce dy dx + d2 f dz 2 + dm+1 s =? dtm+1 You can have only one optional argument, and up to nine arguments in total. The \newcommand{...} definitions should go in the document preamble.

16 How about these? dy dx, d2 f dm+1 s dz 2, etc? dtm+1 You can deal with these by defining a macro with an optional argument: \newcommand{\derivn}[3][]{\frac{d^{#1}#2}{d#3^{#1}}} And then: typing \[\derivn{y}{x}+\derivn[2]{f}{z}+\derivn[m+1]{s}{t}=?\] will produce dy dx + d2 f dz 2 + dm+1 s =? dtm+1 You can have only one optional argument, and up to nine arguments in total. The \newcommand{...} definitions should go in the document preamble.

17 How about these? dy dx, d2 f dm+1 s dz 2, etc? dtm+1 You can deal with these by defining a macro with an optional argument: \newcommand{\derivn}[3][]{\frac{d^{#1}#2}{d#3^{#1}}} And then: typing \[\derivn{y}{x}+\derivn[2]{f}{z}+\derivn[m+1]{s}{t}=?\] will produce dy dx + d2 f dz 2 + dm+1 s =? dtm+1 You can have only one optional argument, and up to nine arguments in total. The \newcommand{...} definitions should go in the document preamble.

18 How about these? dy dx, d2 f dm+1 s dz 2, etc? dtm+1 You can deal with these by defining a macro with an optional argument: \newcommand{\derivn}[3][]{\frac{d^{#1}#2}{d#3^{#1}}} And then: typing \[\derivn{y}{x}+\derivn[2]{f}{z}+\derivn[m+1]{s}{t}=?\] will produce dy dx + d2 f dz 2 + dm+1 s =? dtm+1 You can have only one optional argument, and up to nine arguments in total. The \newcommand{...} definitions should go in the document preamble.

19 One last thing for this part... Which do you prefer: x y z or x y z? If the latter then you can re-new the \le and \ge commands with \renewcommand{\le}{\leqslant} \renewcommand{\ge}{\geqslant}

20 L A TEX environments Environments are a big part of L A TEX. They allow us to switch into a tailored environment, do our thing, and then switch out again. The environment can be designed to suit its purpose. We are going to study theorem-like environments and for this include the following material in your preamble: \usepackage{amsthm} \newtheorem{theorem}{theorem}[section] \newtheorem{lemma}[theorem]{lemma} \newtheorem{proposition}[theorem]{proposition} \newtheorem{corollary}[theorem]{corollary} These will make Theorems, Lemmas, Propositions, Corollaries and proofs available to you.

21 L A TEX environments Environments are a big part of L A TEX. They allow us to switch into a tailored environment, do our thing, and then switch out again. The environment can be designed to suit its purpose. We are going to study theorem-like environments and for this include the following material in your preamble: \usepackage{amsthm} \newtheorem{theorem}{theorem}[section] \newtheorem{lemma}[theorem]{lemma} \newtheorem{proposition}[theorem]{proposition} \newtheorem{corollary}[theorem]{corollary} These will make Theorems, Lemmas, Propositions, Corollaries and proofs available to you.

22 L A TEX environments Environments are a big part of L A TEX. They allow us to switch into a tailored environment, do our thing, and then switch out again. The environment can be designed to suit its purpose. We are going to study theorem-like environments and for this include the following material in your preamble: \usepackage{amsthm} \newtheorem{theorem}{theorem}[section] \newtheorem{lemma}[theorem]{lemma} \newtheorem{proposition}[theorem]{proposition} \newtheorem{corollary}[theorem]{corollary} These will make Theorems, Lemmas, Propositions, Corollaries and proofs available to you.

23 L A TEX environments Environments are a big part of L A TEX. They allow us to switch into a tailored environment, do our thing, and then switch out again. The environment can be designed to suit its purpose. We are going to study theorem-like environments and for this include the following material in your preamble: \usepackage{amsthm} \newtheorem{theorem}{theorem}[section] \newtheorem{lemma}[theorem]{lemma} \newtheorem{proposition}[theorem]{proposition} \newtheorem{corollary}[theorem]{corollary} These will make Theorems, Lemmas, Propositions, Corollaries and proofs available to you.

24 L A TEX environments Note once again the use of \usepackage to load in an additional package: \usepackage{amsthm} This one makes the proof environment available (the others can be used without this package). But what use is a theorem without a proof!

25 L A TEX environments Here s an example using \label and \ref: Theorem 1 (Cauchy-Goursat Theorem) If a function f: C C is single-valued and analytic within and on a closed curve C C then f(z)dz = 0. C Proof of Theorem 1. See, for example, Introduction to Complex Variables and Applications by R.V. Churchill, McGraw-Hill, 1948.

26 Here s the code... \begin{theorem}[cauchy-goursat Theorem]\label{thm:CG-thm} If a function $f\colon\mathbb{c}\to\mathbb{c}$ is single-valued and analytic within and on a closed curve $C\subset\mathbb{C}$ then \[ \int_c f(z)\, dz = 0. \] \end{theorem} \begin{proof}[proof of Theorem~\ref{thm:CG-thm}] See, for example, \textit{introduction to Complex Variables and Applications} by R.V.~Churchill, McGraw-Hill, \end{proof} Let s move to the lab3.tex document for the next part.

27 Here s the code... \begin{theorem}[cauchy-goursat Theorem]\label{thm:CG-thm} If a function $f\colon\mathbb{c}\to\mathbb{c}$ is single-valued and analytic within and on a closed curve $C\subset\mathbb{C}$ then \[ \int_c f(z)\, dz = 0. \] \end{theorem} \begin{proof}[proof of Theorem~\ref{thm:CG-thm}] See, for example, \textit{introduction to Complex Variables and Applications} by R.V.~Churchill, McGraw-Hill, \end{proof} Let s move to the lab3.tex document for the next part.

28 Here s the code... \begin{theorem}[cauchy-goursat Theorem]\label{thm:CG-thm} If a function $f\colon\mathbb{c}\to\mathbb{c}$ is single-valued and analytic within and on a closed curve $C\subset\mathbb{C}$ then \[ \int_c f(z)\, dz = 0. \] \end{theorem} \begin{proof}[proof of Theorem~\ref{thm:CG-thm}] See, for example, \textit{introduction to Complex Variables and Applications} by R.V.~Churchill, McGraw-Hill, \end{proof} Let s move to the lab3.tex document for the next part.

29 Our next task is to learn how to create tables. For example, Left centre centre right L1 C1 C1 R1 L2 C2 C2 R2 Table : A simple table showing left, centre and right alignment, and horizontal and vertical lines. \begin{table}\centering\begin{tabular}{ l c c r }\hline Left & centre & centre & right \\\hline L1 & C1 & C1 & R1 \\ L2 & C2 & C2 & R2 \\\hline \end{tabular}\caption{a simple table showing left, centre and right alignment, and horizontal and vertical lines.\label{tab:simple}}\end{table}

30 Our next task is to learn how to create tables. For example, Left centre centre right L1 C1 C1 R1 L2 C2 C2 R2 Table : A simple table showing left, centre and right alignment, and horizontal and vertical lines. \begin{table}\centering\begin{tabular}{ l c c r }\hline Left & centre & centre & right \\\hline L1 & C1 & C1 & R1 \\ L2 & C2 & C2 & R2 \\\hline \end{tabular}\caption{a simple table showing left, centre and right alignment, and horizontal and vertical lines.\label{tab:simple}}\end{table}

31 Our last topic will give an introduction to tikz. This is a very powerful graphics package for creating plots in L A TEX. It is also BIG and complicated. We will merely scratch the surface... For the sake of demonstration suppose that we want to plot y = 3sin(2x 1) and z = 2 x 2 /2 on the same set of axes with x [ 3,5]. First make sure to load tikz and pgfplots in the preamble: \usepackage{tikz,pgfplots}

32 Here it is... 0 y = 3sin(2x 1) z = 2 x 2 / Figure : Plot of y = 3sin(2x 1) and z = 2 x 2 /2 for x [ 3,5]

33 And here s the code... \begin{figure}[b]\centering\begin{tikzpicture}[scale=0.9] \begin{axis}[grid=major,inner axis line style={=>}, xtick={-3, -2,...,5},legend pos=outer north east] \addplot[blue,mark=dot] expression[domain=-3:5,samples=40]{3*sin(deg(2*x-1))}; \addlegendentry{$y=3\sin(2x-1)$}; \addplot[red,mark=o] expression[domain=-3:5,samples=20]{2-x^2/2}; \addlegendentry{$z=2-x^2/2$}; \end{axis}\end{tikzpicture} \caption{plot of $y=3\sin(2x-1)$ and $z=2-x^2/2$ for $x\in[-3,5]$\label{fig:my-tikz-plot}}\end{figure}

34 And here s the code... \begin{figure}[b]\centering\begin{tikzpicture}[scale=0.9] \begin{axis}[grid=major,inner axis line style={=>}, xtick={-3, -2,...,5},legend pos=outer north east] \addplot[blue,mark=dot] expression[domain=-3:5,samples=40]{3*sin(deg(2*x-1))}; \addlegendentry{$y=3\sin(2x-1)$}; \addplot[red,mark=o] expression[domain=-3:5,samples=20]{2-x^2/2}; \addlegendentry{$z=2-x^2/2$}; \end{axis}\end{tikzpicture} \caption{plot of $y=3\sin(2x-1)$ and $z=2-x^2/2$ for $x\in[-3,5]$\label{fig:my-tikz-plot}}\end{figure}

35 The integral test from lectures The early terms are not significant for a convergence test. So... N+1 m 4 N x+3 dx k=m 4 N k +3 m 1 4 x+3 dx Hence: the series and integral behave the same as N

36 It is clear that this is sophisticated material. There is a lot to take in. These examples will reward study and you can ask in the labs if you get stuck. Or you can look online there is a huge amount of information. Let s just note that the tikz plot is contained in a tikzpicture environment. This itself is within a figure environment which, like the table environment also floats and allows for a labelled and numbered caption. You need to process this with DVI->PS and then PS->PDF to get the tikz graphics.

37 Lab 3 Lab 3 For the next lab session you should download the handwritten material in lab3-proofs-to-create.pdf Spend some time understanding these results they are important and then create your own versions in L A TEX.

Overview (MA2730,2812,2815) lecture 3 End of Lecture Computational and αpplie Mathematics Lab session 3 For the next lab session you should download the

38 Overview (MA2730,2812,2815) lecture 3 End of Lecture Computational and αpplie Mathematics Lab session 3 For the next lab session you should download the handwritten material in lab3-proofs-to-create.pdf Spend some time understanding these results they are important and then create your own versions in LATEX.

L A TEX Lab 3: advanced concepts

L A TEX Lab 3: advanced concepts Simon Shaw November 20, 205 Lab 3: macros Suppose that you wanted to write many instances of dy dx. In code this is \frac{dy}{dx} But you might still think it is a lot