Typesetting Mathematics with LaTeX - Day 2

SU
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Recall Symbols AdEqn Greek Letters
Typesetting mathematics with LATEX
Suddhasheel Ghosh
Department of Civil Engineering,
Jawaharlal Nehru Engineering College
Aurangabad, Maharashtra
431003
LATEX for Technical and Scientiﬁc Documents
January 12 - 16, 2015
Day II
shudh@JNEC LATEX@ JNEC 1 / 30

Outline
1 What did we learn about typesetting Math
2 Mathematics Symbology
3 Advanced equations
4 Greek Letters

In this workshop:
Day 1: Introduction to LATEX
Day 2: Typesetting Mathematics
Day 3: Writing your thesis / technical report
Day 4: Make your presentations
Day 5: Basics of science communication

Outline
4 Greek Letters

Playing with mathematics text
Inline and display math
There are two kinds of math used in LATEX.
Inline math (surrounded by $ signs)
Display math (surrounded by square brackets [ and ] or $$
About any point x in a metric space M we deﬁne the open ball of
radius r > 0 about x as the set
B(x;r) = {y ∈ M : d(x,y) < r}.
About any point $x$ in a metric space $M$ we define the
open ball of radius $r > 0$ about $x$ as the set
[
B(x;r) = {y in M : d(x,y) < r}.
]

SU
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Typesetting equations
Numbered equations
Numbered equations can be typeset by using
begin{equation}
R(x,y) = frac{Ax^3+Bx^2+Cx+D}{Ey^3 + Fy + G}
end{equation}
R(x,y) =
Ax3
+Bx2
+Cx+D
Ey3 +Fy +G
(1)

Outline
4 Greek Letters

Deﬁnition symbols
An example of a Cauchy sequence
A sequence of numbers 〈xi〉n
i=1
, is called a Cauchy sequence, if
∃ > 0, ∀N ∈ N, such that for all m,n ∈ N,
|xm −xn| <
A sequence of numbers $leftlangle x_i
rightrangle_{i=1}^n$,
is called a Cauchy sequence, if $exists,epsilon > 0$,
$forall Ninmathbb{N}$, such that for all
$m,ninmathbb{N}$,
[
leftvert x_m - x_n rightvert < epsilon
]

Deﬁnition symbols
Continuous function example
We can say that
lim
x→c
f (x) = L
if ∀ε > 0 , ∃δ > 0 such that ∀x ∈ D that satisfy 0 < |x−c| < δ , the
inequality |f (x)−L| < ε holds.
We can say that
[
limlimits_{xrightarrow c} f(x) = L
]
if $forall,varepsilon > 0$ , $exists, delta > 0$
such that $forall xin D$ that satisfy $0 < | x - c | <
delta$ ,
the inequality $|f(x) - L| < varepsilon$ holds.

Outline
4 Greek Letters

Multiline equations
... and alignment
If h ≤ 1
2 |ζ−z| then
|ζ−z−h| ≥
1
2
|ζ−z|
and hence
1
ζ−z−h
−
1
ζ−z
=
(ζ−z)−(ζ−z−h)
(ζ−z−h)(ζ−z)
=
h
(ζ−z−h)(ζ−z)
≤
2|h|
|ζ−z|2
. (2)
If $h leq frac{1}{2} |zeta - z|$ then
[ |zeta - z - h| geq frac{1}{2}
|zeta - z|]
and hence
begin{eqnarray}
left| frac{1}{zeta - z - h} -
frac{1}{zeta - z} right|
& = & left|
frac{(zeta - z) -
(zeta - z - h)}{(zeta - z - h)(zeta -
z)}
right| nonumber & = &
left| frac{h}{(zeta - z - h)(zeta -
z)}
right| nonumber
& leq & frac{2 |h|}{|zeta - z|^2}.
end{eqnarray}

Case based deﬁnitions
An example of a continuous, nowhere differential function is given
as
f (x) =
0 x is irrational
1 x is rational
An example of a continuous, nowhere differential function
is given as
[
f(x) =
begin{cases}
0 &{x mathrm{ is irrational}}
1 &{x mathrm{ is rational}}
end{cases}
]

Case based deﬁnitions
... a more professional one
An example of a continuous, nowhere differential function is given
as
f (x) =
1 x ∈ Q
0 x ∈ R−Q
An example of a continuous, nowhere differential function
is given as
[
f(x) =
begin{cases}
1 &{xinmathbb{Q}}
0 &{xinmathbb{R}-mathbb{Q}}
end{cases}
]

Fractions and binomials
To typeset:
n!
k!(n−k)!
=
n
k
Use
[
frac{n!}{k!(n-k)!} = binom{n}{k}
]
For the computation of a permutation n
Pk, we would use the
following formula:
n
Pk = (n−1)(n−2)...(n−k +1)
Exactly k factors
[
{}^n P_k = underbrace{(n-1)(n-2)
dotsc(n-k+1)}_{mbox{Exactly $k$ factors}}
]

What about a continued fraction?
Any idea?
To get,
x = a0 +
1
a1 +
1
a2 +
1
a3 +
1
a4
[
x = a_0 + cfrac{1}{a_1
+ cfrac{1}{a_2
+ cfrac{1}{a_3 + cfrac{1}{a_4} } } }
]

Square roots
... and other roots too!
A = 4x3 +3x2 −5x+1
B =
3
4x3 +3x2 −5x+1
[
A = sqrt{4x^3 + 3x^2 - 5x + 1}
]
[
B = sqrt[3]{4x^3 + 3x^2 - 5x + 1}
]

Square roots
...use in mathematical expressions
Given a quadratic equation ax2
+bx+c = 0, the roots of the
equation are given by the following formula:
x =
−b± b2 −4ac
2a
Given a quadratic equation $ax^2 + bx + c = 0$, the roots
of the equation are given by the following formula:
[
x = frac{-b pm sqrt{b^2 - 4ac} }{2a}
]

Sums and Products
n
i=0
ai = a1 +a2 +a3 +···+an
[
sum_{i=0}^n a_i = a_1 + a_2 + a_3 + dots + a_n
]
n
i=0
ai = a1 ·a2 ·a3 ·····an
[
prod_{i=0}^n a_i = a_1 cdot a_2 cdot a_3
cdot dots cdot a_n
]

Integrals and Differentials
∞
0
e−x2
dx =
π
2
[
int_0^infty e^{-x^2} dx=frac{sqrt{pi}}{2}
]
If f (x,y) = x2
+y2
, then:
∂f
∂x
= 2x
If $f(x,y) = x^2 + y^2$, then:
[
frac{partial f}{partial x} = 2x
]

Mathematical operators
Integration examples
In the ﬁeld of integral calculus, it is known that
sinx = −cosx
and
cosx = sinx
In the field of integral calculus, it is known that
[
intsin x = -cos x
]
and
[
intcos x = sin x
]

Mathematical operators and notation
Fourier transform
The Fourier transform, for a function f (x), is given as
ˆf (ξ) =
∞
−∞
f (x) e−2πixξ
dx, ∀ξ ∈ R
and the inverse Fourier transform is given as,
f (x) =
∞
−∞
ˆf (ξ) e2πixξ
dξ ∀x ∈ R
The Fourier transform, for a function $f(x)$, is given as
[
hat{f}(xi) = int_{-infty}^{infty} f(x) e^{-2 pi i
x xi}, dx, , qquad forall xiinmathbb{R}
]
and the inverse Fourier transform is given as,
[
f(x) = int_{-infty}^{infty} hat{f}(xi) e^{2 pi i x
xi}, dxi , qquad forall x inmathbb{R}
]

Mathematical Operators
Continuity equation ... and vector notation
Mathematically, the integral form of the continuity equation is:
dq
dt
+
S
j·dS = Σ
which in vector notation can also be written as
dq
dt
+
S
j ·dS = Σ
Mathematically, the integral form of the continuity
equation is:
[
frac{d q}{d t} + oiint_Smathbf{j} cdot
dmathbf{S} = Sigma
]
which in vector notation can also be written as
[
frac{d q}{d t} + oiint_Svec{j} cdot dvec{S} =
Sigma
]

Typesetting Matrices
... and determinants
The determinant of the matrix A, where
A =
1 2 3
4 5 6
7 8 9
is zero.
[
A =
begin{vmatrix}
1 &2 &3
4 &5 &6
7 &8 &9
end{vmatrix}
]

Typesetting Matrices
Typically some long ones
The Gaussian Elimination Method can be used to ﬁnd the solution
of the following system of equations, represented in the matrix
form: 





a11 a12 a13 ... a1n
a21 a22 a23 ... a2n
...
...
...
...
...
an1 an2 an3 ... ann












x1
x2
...
xn






=






b1
b2
...
bn






[
begin{bmatrix}
a_{11} & a_{12} &a_{13} &dots &a_{1n} a_{21} & a_{22}
&a_{23} &dots &a_{2n}
vdots &vdots &vdots &ddots &vdots a_{n1} & a_{n2}
&a_{n3} &dots &a_{nn}
end{bmatrix}
begin{bmatrix}
x_1 x_2 vdots x_n
end{bmatrix}
=
begin{bmatrix}
b_1 b_2 vdots b_n
end{bmatrix}
]

Array based typesetting of equations
Least squares example
The least squares method is used to ﬁnd the solution of a system of
n equations, with m variables where m > n. The system of
equations is given as follows:
a11x1 +a12x2 +... +a1mxm = b1
a21x1 +a22x2 +... +a2mxm = b2
... ... ... ... ... ...
an1x1 +an2x2 +... +anmxm = bm
(3)
The least squares method is used to find the solution of
a system of $n$ equations, with $m$ variables where
$m > n$. The system of equations is given as follows:
begin{equation}
begin{array}{llllll}
a_{11} x_1 &+a_{12} x_2 &+dotsc &+a_{1m} x_m &= &b_1
a_{21} x_1 &+a_{22} x_2 &+dotsc &+a_{2m} x_m &= &b_2
dots &dots &dots &dots &dots &dots
a_{n1} x_1 &+a_{n2} x_2 &+dotsc &+a_{nm} x_m &= &b_m
end{array}
end{equation}

Array based typesetting of equations
Least squares example ... with a twist
The least squares method is used to ﬁnd the solution of a system of
n equations, with m variables where m > n. The system of
equations is given as follows:
Solve



a11x1 +a12x2 +... +a1mxm = b1
a21x1 +a22x2 +... +a2mxm = b2
... ... ... ... ... ...
an1x1 +an2x2 +... +anmxm = bm
(4)
The least squares method is used to find the solution of
a system of $n$ equations, with $m$ variables where
$m > n$. The system of equations is given as follows:
begin{equation}
rotatebox{90}{Solve} left{ begin{array}{llllll}
a_{11} x_1 &+a_{12} x_2 &+dotsc &+a_{1m} x_m &= &b_1
a_{21} x_1 &+a_{22} x_2 &+dotsc &+a_{2m} x_m &= &b_2
dots &dots &dots &dots &dots &dots
a_{n1} x_1 &+a_{n2} x_2 &+dotsc &+a_{nm} x_m &= &b_m
end{array}
right.
end{equation}

Outline
4 Greek Letters

SU
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ASH
EEL.CO
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Greek letters
The small greek letters are
α,β,γ,δ, ,ε,ζ,η,θ,ϑ,κ, ,λ,µ,ν,ξ,π, ,ρ, ,σ,ς,τ,υ,φ,ϕ,χ,ψ,ω,
and the capital greek letters are given as
A,B,Γ,∆,E,Z,H,Θ,K,Λ,M,N,Ξ,Π,P,Σ,T,Υ,Φ,X,Ψ,Ω
The small greek letters are $alpha, beta, gamma,
delta, epsilon, varepsilon, zeta, eta, theta,
vartheta, kappa, varkappa, lambda, mu, nu, xi,
pi, varpi, rho, varrho, sigma, varsigma, tau,
upsilon, phi, varphi, chi, psi, omega$, and
the capital greek letters are given as $ A, B, Gamma,
Delta, E, Z, H, Theta, K, Lambda, M, N, Xi, Pi, P,
Sigma, T, Upsilon, Phi, X, Psi, Omega $

Example with greek letters
... and something more
If Z1,...,Zk are independent, standard normal
random variables, then the sum of their squares,
Q =
k
i=1
Z2
i ,
is distributed according to the ξ2
distribution
with k degrees of freedom. This is usually
denoted as
Q ∼ χ2
(k) or Q ∼ χ2
k.
The chi-squared distribution has one
parameter: k – a positive integer that speciﬁes
the number of degrees of freedom (i.e. the
number of Zi’s)
If $Z_1, dots, Z_k$ are independent,
standard normal random variables,
then the sum of their squares,
[
Q = sum_{i=1}^k Z_i^2 ,
]
is distributed according to the $xi^2$
distribution with $k$ degrees of freedom.
This is usually denoted as
[
Q sim chi^2(k) text{or} Q
sim chi^2_k .
]
The chi-squared distribution has one
parameter: $k$ -- a positive integer that
specifies the number of degrees
of freedom (i.e. the number of $Z_i$’s)

In the upcoming sessions
Typing a thesis / report / dissertation
Table of contents / ﬁgure / tables
Managing Bibliography
Cross referencing
Paper referencing
Style ﬁles for theses and bibliography

Typesetting Mathematics with LaTeX - Day 2

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