Just Enough L A TEX, Week 4

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1 Just Enough L A TEX, Week Department of Mathematics Michigan State University East Lansing, MI USA weil@math.msu.edu October 24, 2008

2 Typesetting Fractions The basic command to produce a fraction is \frac{numerator}{denominator} which can only be used in math mode. In a case such as $\frac{x^2}{z^e}$ the output x 2 z e can be difficult to read. To use a larger font size, the one used in display math mode, type instead $\dfrac{x^2}{z^e}$, which produces x 2 z e which is the same as in display mode, $$\frac{x^2}{z^e}$$. To produce the smaller version in display mode type $$\tfrac{x^2}{z^e}$$. Compare x 2 z e to x 2 z e.

3 Typesetting Binomial Coefficients Recall the conclusion of the Binomial Theorem. n ( ) n (b + a) n = b k a n k k k=1 The so-called binomial coefficients are produced in a fashion analogous to fractions. Specifically $\binom{m}{n}$ ( ) produces ( m ) m n while $\dbinom{m}{n}$ produces. Likewise in display ( ) n m mode $$\binom{m}{n}$$ produces and n $$\tbinom{m}{n}$$ produces ( ) m n.

4 Modular Equivalence L A TEX provides four different way to express that x and y are equivalent modulo a positive integer n; that is, y x = kn for some integer (positive or negative) k. They are presented in the following table. Typing Produces x\equiv y \mod{n} x y mod n x\equiv y \bmod{n} x y mod n x\equiv y \pmod{n} x y (mod n) x\equiv y \pod{n} x y (n)

5 Typesetting Roots Square roots are produced with the command \sqrt{expression}. For example typing $\sqrt{x^3-8}$ produces x 3 8 for math mode in line and typing $$\sqrt{x^3-8}$$ produces x 3 8 in display math mode. Note the extra space at the top of the expression. An optional argument for the command produces general roots. For example typing $\sqrt[7]{x^3-8}$ produces 7 x 3 8 for math mode in line and typing $$\sqrt[7]{x^3-8}$$ produces 7 x 3 8 in display math mode.

6 Fonts in Math Mode Standard L A TEX provides the calligraphic font in math mode (upper case only) that are produced with the command \mathcal. Some examples are A, B, C, D, E, F produced with the command $\mathcal{a, B, C, D, E, F}$. The amssymb package provides the Blackboard Bold fonts and the Fraktur fonts in math mode as well as numerous mathematical symbols. The Blackboard Bold fonts are only upper case. They are produced with the command \mathbb. For example $\mathbb{n, R, C, Z}$ produces N, R, C, Z. The Fraktur fonts, available in both upper case and lower case, are produced with the command \mathfrak; e.g., A, B, C, D, E, a, b, c is produced by $\mathfrak{a, B, C, D, a, b, c}$.

7 Stating Assertions with the amsthm Package Including \usepackage{amsthm} in the preamble permits the creation of new environments that produce assertions such as definitions, lemmas, propositions, theorems, and remarks etc. with either of two commands \newtheorem or \newtheorem*. Each has two mandatory arguments and the first has two optional ones as well. Both commands must occur in the preamble. For example the command \newtheorem{theorem}{theorem} creates an environment theorem whose heading is Theorem followed by a number identifying the theorem; 1 for the first use, 2 for the second etc. The numbers are assigned from a counter named theorem.

8 A Simple Example For example typing \begin{theorem}\label{th1} This is the first theorem. \end{theorem} \begin{theorem}\label{th2} This is the second theorem. \end{theorem} produces Theorem 1. This is the first theorem. Theorem 2. This is the second theorem.

9 A Numbering Option In the case of a long article containing numerous theorems, numbering them consecutively can become unruly. In such a situation numbering them according to section is an attractive alternative. To do so add the option [section] to the defining command. Specifically \newtheorem{theorem}{theorem}[section]. (The position of this option is important.) Assuming the theorems are being created in the fourth section the outcome would be Theorem 4.1. This is the first theorem in Section 4. Theorem 4.2. This is the second theorem in Section 4.

10 Additional Types of Assertions To include lemmas as well as theorems in the preamble type \newtheorem{lemma}{lemma}. Then typing \begin{lemma}\label{lem1} This is the first lemma. \end{lemma} produces Lemma 1. This is the first lemma. Of course this particular \newtheorem command produces a counter named lemma.

11 Numbering Consecutively with Theorems Using the option [theorem] causes lemmas to be numbered using the same counter that numbers theorems; specifically \newtheorem{lemma}[theorem]{lemma}. (Note the different location of the option.) Then what might be seen is Lemma 1. This is the first lemma. Theorem 2. This is the first theorem. Theorem 3. This is the second theorem. If theorems are numbered according to section number, the same would be true of lemmas. Additional assertions such as propositions and corollaries may also be numbered using the same counter as is used by theorem. With an optional argument in the position of this one, no new counter is created.

12 Other Styles The amsthm package also provides two additional theorem styles besides the default one already exhibited. They are definition and remark. They are changed in the preamble by typing \theoremstyle{style} before the \newtheorem command that creates the assertion type. For example typing in the preamble \theoremstyle{definition} \newtheorem{definition}{definition} \theoremstyle{remark} \newtheorem{remark}{remark} permits Definition 1. This is the first definition. Remark 1. This is the first remark.

13 Named Assertions The preamble command \newtheorem* is used to produce assertions that are named rather than numbered. This command permits only two mandatory arguments. For example the command \newtheorem*{wo}{well Ordering Principle} allows the typing of \begin{wo} Every non-empty set can be well-ordered. \end{wo} to produce Well Ordering Principle. Every non-empty set can be well-ordered.

14 An Option for All Assertion Environments All of the environments created with either of the preamble commands \newtheorem or \newtheorem* have an optional argument that can be used for inserting additional information about the assertion before its statement. For example, typing \begin{lemma}[see \cite{t}] If $f$ is differentiable at $x$, then $f$ is continuous at $x$. \end{lemma} will produce Lemma 2. (see [7]) If f is differentiable at x, then f is continuous at x. assuming that the famous text book by George Thomas is the seventh item in the bibliography and that the default method of identifying bibliographic references is used.

15 The amsthm Package s Proof Environment After stating an assertion requiring a proof, type \begin{proof}. Flush with the left margin the text Proof. is produced. If say \begin{proof}[proof of Theorem 1] it typed, then Proof of Theorem 1. is produced. At the end of the proof type \end{proof} and the end of proof symbol appears flush to the right margin. If the proof ends with a displayed formula the appears at the end of the next line. It s possible, with the command \qedhere to place the at the end of the displayed formula instead. How this is done is explained in the next frame.

16 Single Line Displayed Formulas The easiest method to produce single line, unidentified, displayed formulas is with the formula bracketed by $$ and $$. As a simple example (a+b) 3 = a 3 + 3a 2 b + 3ab 2 + b 3 is produced by typing $$ (a+b)^3=a^3+3a^2b+ab^2+b^3 $$ To number such a formula requires the amsmath package. Then type the formula inside the equation environment. That is, \begin{equation}\label{eq1} (a+b)^3= a^3+3a^2b+3ab^2+b^3 \end{equation} produces (a+b) 3 = a 3 + 3a 2 b + 3ab 2 + b 3 (1) Using equation* omits the number, but permits the use of \qedhere which effectively replaces the number with.

17 Multi-Lined Displayed Formulas with align In any multi-lined formula or expression, the different lines are separated by \\. The align environment allows the different lines to be aligned at a point selected by the user with an &. Each line is given a number. For example \begin{align} \sin(x+\frac{\pi}{2})&=\sin x\cos\frac{\pi}{2}+ \cos x\sin\frac{\pi}{2}\\ &=\cos x \end{align} produces sin(x + π 2 ) = sin x cos π 2 + cos x sin π 2 (2) = cos x (3) To eliminate the numbers, replace align with align*.

18 Multi-Lined Formula with One Number Numbering every line of a multi-lined formula is often excessive. The number at the end of any line may be eliminated with the command \notag before the \\ command. An often better alternative is to combine the equation environment with the split environment. For example typing \begin{equation}\label{eq4} \begin{split} \sin(x+\frac{\pi}{2})&=\sin x\cos\frac{\pi}{2}+ \cos x\sin\frac{\pi}{2}\\ &=\cos x \end{split} \end{equation} produces sin(x + π 2 ) = sin x cos π 2 + cos x sin π 2 = cos x (4)

19 Locating and Changing the Tags By choosing the option [tbtags] for the amsmath package the number in the preceding example is moved to the bottom of the displayed formula (and flush to the right margin), or if, in addition the option [leqno] has been selected, the number will appear at the top flush with the left margin. Equations, like assertions, may be numbered according to the section in which they appear with the preamble command \numberwithin{equation}{section} The user may override the number tags with a personal choice using the command \tag{label} or \tag*{label}. These commands are placed before the \\ command. The first results in the user s choice surrounded by parentheses while the second omits the parentheses.

20 Other Multi-Lined Environments The gather and gather* environments center all lines and, in the case of gather, numbers each line. The multline and multline* environments place the first line flush to the left margin, place the last line flush to the right margin and centers all other lines. In the case of multline a number appears after the last line aligned to the right margin unless the class option leqno has been selected in which case the number is placed before the first line flush with the left margin. The alignment character & must not appear in any line of any of these environments.

21 Two or More Columns of Equations The align and align* environments may be used to align the equations (or expressions) in two or more columns of equations (or expressions). The first & aligns the equations in the first column and the second & separates the first and second columns while the next & aligns the equations in the second column etc. For example typing \begin{align*} 6&\equiv 0\bmod{6}&7&\equiv 1\bmod{6}\\ 8&\equiv 2\bmod{6}&9&\equiv 3\bmod{6}\\ 10&\equiv 4\bmod{6}&11&\equiv 5\bmod{6} \end{align*} produces 6 0 mod mod mod mod mod mod 6

22 Alternate Numbering When the expressions are closely related as in the previous example, it s convenient to have a method of numbering them to reflect that fact. The subequations environment provides this method. To produce type 6 0 mod mod 6 (5a) 8 2 mod mod 6 (5b) 10 4 mod mod 6 (5c) \begin{subequations} \begin{align} 6&\equiv 0\bmod{6}&7&\equiv 1\bmod{6}\\ 8&\equiv 2\bmod{6}&9&\equiv 3\bmod{6}\\ 10&\equiv 4\bmod{6}&11&\equiv 5\bmod{6} \end{align} \end{subequations}

23 The \intertext Command The command\intertext{text to insert} is used to insert a line or more of text between lines of any of the multi-lined environments introduced. For example to produce and type \begin{align*} f(x)&= x^2\\ \intertext{and} g(x)&=6x-7 \end{align*} f (x) = x 2 g(x) = 6x 7

24 Spacing Hints When aligning at a relationship such as =, <, etc. put the & symbol before the relationship symbol for the best spacing. When aligning on a relationship and a line must be broken before the next relationship, break and align on a + or, but put extra space after the &. For example to produce (a + b + c) (A + B + C ) = aa + ab + ac + ba + bb + bc type + ca + cb + cc \begin{align*} (a+b+c)\times(a+b+c)&=aa+ab+ac+ba+bb+bc\\ &\phantom{=}+ca+cb+cc \end{align*}

25 The aligned and gathered Environments Unlike the previous environments, these two can be used in line and hence must be preceded with a begin math mode of some type. The formatting rules for each of these environments are the same as those of their namesakes. For example The pair u xx + u yy = 0 for x 2 + y 2 < 1 u(x,y) = f (x,y) for x 2 + y 2 is a boundary value = 1 problem. is produced by typing The pair $\begin{aligned} u_{xx}+u_{yx}&=o\text{ for }x^2+y^2<1\\ u(x,y)&=0\f(x,y)\text{ for }x^2+y^2=1 \end{aligned}$ is a boundary value problem. The command \text is used as a convenient way to insert text while in math mode. Note that the extra space needed to separate x from the text is with a space in the argument of \text where a space is recognized.

26 The cases Environment To produce the type of structure { 0 if x is a rational number f (x) = 1 if x is an irrational number (6) before the appearance of the cases environment required the array environment and many extra steps. Now it s much easier. Simply type $$f(x)=\begin{cases} 0&\text{if } x \text{ is a rational number}\\ 1&\text{if } x \text{ is an irrational number} \end{cases}$$ The cases environment can be used in line in the same way as the aligned environment.

LaTeX Seminar III: Environments and More Advanced Mathematical Typesetting

LaTeX Seminar III: Environments and More Advanced Mathematical Typesetting Clifford E. Weil March 24, 2004 1 General Environments We have already encountered two environments. They are the document environment