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Arnav's Notes

Calculus & its Applications - Concise Notes 1

MATH40004

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Colour Code - Definition are green in these notes, Consequences are red and Causes are blue

Chapter 1, Limits of Functions, continuity

Definition 1. $\epsilon - \delta$ Definition of Limit


Let $f$ be a function defined at all points near $X_0$, except possible at $x_0$
let $l$ be a real number.
We say that $l$ is a limit of $f(x)$ as $x$ approaches $x_0$ if

\[\forall \epsilon >0\ \exists\delta>0 \text{ s.t } |f(x)-l|<\epsilon \text{ for } |x-x_0|<\delta, x\neq x_0\]

We write $lim_{x\rightarrow x_0}f(x)=l$

Basic Properties of Limits

  1. Sum rule
    $\lim_{x \to x_0}[f(x)+g(x)]=\lim_{x \to x{0}} f(x) + \lim_{x \to x{0}} g(x)$

  2. Product rule
    $\lim_{x \to x{0}} [f(x)g(x)]=\lim_{x \to x{0}} f(x)\lim_{x \to x{0}} g(x)$

  3. Reciprocal rule
    if $\lim{x \to x_{0}} f(x)\neq 0$ then
    $\lim_{x \to x{0}} [1/f(x)] = 1/\lim_{x \to x{0}} f(x)$

  4. Quotient rule
    If $\lim_{x \to x{0}} g(x)\neq 0$ then
    $\lim_{x \to x{0}} [f(x)/g(x)] = \lim_{x \to x{0}} f(x)/\lim_{x \to x{0}} g(x)$

  5. Composite function rule
    If $h(x)$ is conitnuous at $\lim_{x \to x{0}} f(x)$ then
    $\lim_{x \to x{0}} h(f(x)) = h(\lim_{x \to x{0}} f(x))$

Definition 2. The $\epsilon - A$ definition of $\lim{x \to \infty} f(x)=l$

Let $f(x)$ be defined on a domain containing, the interval $(a,\infty)$.A real number $l$is the limit of $f(x)$ as $x$ approaches $\infty$ if for every $\epsilon>0$ there exists a $A>a$, such that $|f(x)-l|<\epsilon$ whenver $x>A$. We write $\lim{x \to \infty} f(x)=l$

Definition 3. $\epsilon - B$ definition of $\lim_{x \to x{0}} f(x)=\infty$

Let $f(x)$ be a function defined in an interval containing $x_0$ except possibly at $x=x_0$. We saythat $f(x)$ approaches $\infty$ as $x$ approaches $x_0$, if given any real number $B>0$, there exists $\epsilon >0$, so that whenever $|x-x_0|<\epsilon$ and $x\neq x_0$, we have $f(x)>B$. We write $\lim_{x \to x{0}} f(x)=\infty$

Definition 4. One-Sided limit

Let $f(x)$ be defined for all $x$ in an interval $(x_0,a)$. We say that $f(x)$ approaches $l$ as $x$ approaches $x_0$ from the right if for any $\epsilon>0$ there exists $\delta >0$, such that for all $x_0 <x<x_0 +\delta$ we have $|f(x)-l|<\epsilon$. We Write $\lim_{x \to x{0}-} f(x)=l$

Comparison Test for Limits

  1. $\lim_{x \to x{0}} f(x)=0$ and $|g(x)\leq |f(x)|$ for all $x$ near $x_0$ with $x\neq x_0$, then $\lim_{x \to x{0}} g(x)=0$

  2. $\lim_{x \to \infty} f(x)=0$ and $|g(x)|\leq |f(x)|$ for all large enough $x$ then$\lim_{x \to \infty} g(x)=0$

Two Basic Trigonometric Limits

$\lim_{h \to 0}\frac{sinh}{h}=1$ $\lim_{h\to 0}\frac{cosh-1}{h}=0$

Definition 5. Continuity

We say that $f$ is continuous at $x_0$ if $\lim_{h \to 0}f(x_0+h) = f(x_0)$ Equivalently $\lim_{x \to x{0}} f(x)=f(x_0)$ A totally equivalent definition is: $f(x)$ is continuous at a point $x_0$ if for every $\epsilon>0$ there exists a number $\delta >0$ such that $|f(x)-f(x_0)|<\epsilon$ for all $x$ in the domain of $f$ for which $|x-x_0|<\delta$

Differentiation

Definition with limits

Definition 6. Differentiability

The function $f(x)$ is differentiable at x if "Newton’s quotient"

$\lim{h \to 0}\frac{f(x+h)-f(x)}{h}$ exists. We call this $f’(x)$ the derivative of $f$ at point $x$

Polynomials

Theorem 1.

Let $n$ be an integer $\geq 1$ and let $f(x)=x^n$. Then $f’(x)=\frac{df}{dx}=nx^{n-1}$

Theorem 2.

Let $f(x)=x^a$ where $a$ is any real number and $x>0$. Then $f’(x) = ax^{a-1}$.

General rules, chain rule, rate of changes

General rules

  1. If $c$ is a constant $(cf)’(x)=cf’(x)$

  2. if $f(x),g(x)$ are given functions and $f’(x),g’(x)$ exist, then

    $(f+g)’(x)=f’(x)+g’(x)$

  3. $(fg)’(x)=f’(x)g(x)+f(x)g’(x)$

  4. Let g(x) be a function that has a derivative $g’(x)$ and such that $g(x)\neq0$
    Then $\frac{d}{dx}(\frac{1}{g(x)})=-\frac{g’(x)}{(g(x))^2}$

  5. $\frac{d}{dx}\frac{f(x)}{g(x)}=\frac{g(x)f’(x)-f(x)g’(x)}{(g(x))^2}$

The Chain Rule

\[\frac{d}{dx}(f\circ g)(x)=(f\circ g)'(x)=f'(g(x))g'(x)\]

Theorem 3.

If $f(x)$ is differentiable at $x=x_0$ then it is also continuous there.

Not much notable here. You can prove the derivative of polynomials with fractional powers using implicit differentiation.

Mean Value and Intermediate Value Theorems

For a function $f(x)$ which is defined at a point $c$, we say that $c$ is maximum of $f$ if

$f(c)\geq f(x) \forall x$ where $f$ is defined

The minimum is obvious

Theorem 4.

Let $f$ be a function which is defined and differentiable on the open interval $(a,b)$. Let $c$ be a number in the interval which is a maximum for the function. Then $f’(c)=0, f’(c)=0$ also if $c$ is a minimum of $f$

Theorem 5.

Let $f(x)$ be continuous on the close interval [a,b]. Then $f(x)$ has a maximum and a minimum on this interval.

Theorem 6.

Let $f(x)$ be continuous over the closed interval$a\leq x \leq b$ and differentiable on the interval $a<x<b$. Assume also that $f(a)=f(b)=0$. Then there exists a point $c, a<c<b$ such that $f’(c)=0$

Theorem 7.

Suppose $f$ is continuous on $[a,b]$ and differentiable on $a,b$ Then there exists $a<c<b$ such that $f’(c) = \frac{f(b)-f(a)}{b-a}$

Definition 7.

We say that $f$ is increasing over a given interval if given $x_1,x_2$ in the interval with $x_1\leq x_2$, we have $f(x_1)\leq f(x_2)$ If it is strictly increasing it is the same with $<$ insteaf of $\leq$ Same for decreasing and strictly decreasing

Definition 8.

Let $f(x)$ be continuous in some interval, and differentiable there(even possible at the end points.)
If $f’(x)=0$ in the interval(except possible at endpoints) then $f$ is constant
If $f’(x)>0$ in the interval(except possible at endpoints) then $f$ is strictly increasing
If $f’(x)<0$ in the interval(except possible at endpoints) then $f$ is striclty decreasing\

Theorem 8. Intermediate value theorem

Let $f$ be continuous on the close interval $a\leq x\leq b$. Given any number $y$ between $f(a) and f(b)$, there exists a point $x$ between a and b such that $f(x)=y$

Inverse Functions

Definition 9.

Let $y=f(x)$ be defined on some interval. Given any $y_0$ in the range of $f$, if we can find a unique value $x_0$ in its domain such that $f(x_0)=y_0$, then we an define the inverse funtion $x=g(y)$(sometimes written $x=f^{-1}(y)$

Theorem 9.

Let $f(x)$ be strictly increasing or strictly decreasing. Then the inverse function exists.

Theorem 10.

If $f(x)$ is continuous $[a,b]$ and is strictly increasing(or decreasing), and $f(a)=y_a$ and $f(b)=y_b$, then $x=g(y)$ is defined on $[y_a,y_b]$\

Derivative of inverse functions

Theorem 11.

let $f(x)$ be differentiable on $(a,b)$ and $f’(x)>0$ or $f’(x)<0$ for all $x$ in $(a,b)$. Then the inverse function exists and we have

$g’(y)=\frac{d}{dy}f^{-1}(y)=\frac{1}{f’(x)}$

Exponentials and Logarithms

Geometrical Definition, Derivative

Definition 10.

The quantity $log(x)$ is the area under the curse $\frac{1}{x}$ between $1$ and $x$ if $x\geq 1$ and the negative the area under the curve $\frac{1}{x}$ between $1$ and $x$ if $x$ is in the interval $(0,1)$. In particular $log(1)=0$

Theorem 12.

$log(x)$ is differentiable and $\frac{d}{dx}log(x)=\frac{1}{x}$

Theorem 13.

If $a,b>0$, then $log(ab)=log(a)+log(b)$

Theorem 14.

$log(x)$ is strictly increasing for all $x>0$. Its range is $(-\infty,\infty)$

Theorem 15.

If $n$ is an integer(positive or negative) then $log(a^n)=nlog(a)$ for all $a>0$

Exponential as Inverse of $logx$

Theorem 16.

If $x_1,x_2$ are two numbers then $exp(x_1+x_2)=exp(x_1)exp(x_2)$

Theorem 17.

$exp(x)$is differentiable and $\frac{d}{dx}exp(x)=exp(x)$

Theorem 18.

$\frac{d}{dx}a^x=a^x(log(a))$

Corollary:
$\lim_{h\to 0}\frac{a^h-1}{h}=log(a)$ for $a>0$

Theorem 19.

Let $a$ be any real number and let $f(x) = x^a$ for $x>0$.Then $f’(x)$ exists and $f’(x)=ax^{a-1}$

Function estimates for Small and Large Arguments

Theorem 20. Let $a$ be any real number. Then $\frac{(1+a)^n}{n}\to \infty$ as $n \to \infty$


Corollary:$\frac{e^n}{n}\to \infty$ as $n\to \infty$ since $e=1+a$ for some $a>0$\

Theorem 21.

The function $f(x)=\frac{e^x}{x}$ is strictly increasing for $x>1$ and $\lim_{x\to \infty}f(x)=\infty$
Corollary
The function $x-log(x)$ becomes arbitrarily large as $x$ becomes arbitrarily large. x beats log.
Corrolary
The function $\frac{x}{log(x)}$ becomes large as $x$ becomes large. x beats log
 Corolary
As x becomes large $x^{1/x}$ approches the limit 1.

Theorem 22. exp(x) beats any power of x

Let $m$ be a positive integer. Then the function $f(x)=\frac{e^x}{x^m}$ is strictly increasing for x>m and becomes arbitrarily large as $x$ becomes arbitrarily large.

Logarithmic Differentiation

not much here

L’Hopital’s Rule

Theorem 23.

If $f,g$ are differnetiable on an open interval containing $x_0$, $g(x_0) =f(x_0)=0$, and $g’(x_0)\neq 0$, then
$\lim_{x\to x_0}\frac{f(x)}{g(x)}=\frac{f’(x_0)}{g’(x_0)}$

Theorem 24.

Let $f(x)$ and $g(x)$ be a differentiable on an open interval containing $x_0$(except possible at $x_0$). Assume that $g(x)\neq0$ and $g’(x)\neq 0$ for $x$ in an interval about $x_0$ but with $x\neq x_0$. Assume also that $f.g$ are continous at $x_0$ with $f(x_0)=g(x_0)=0$ and $\lim_{x\to x_0}\frac{f(x)}{g(x)}=l$. Then also:

$\lim_{x \to x_0}\frac{f(x)}{g(x)}=l$

Theorem 25. L’Hopital’s Rule-general case

To find $lim{x\to x_0}\frac{f(x)}{g(x)}$ when $lim{x\to x_0}f(x)$ and $\lim{x\to x_0}g(x)$ are both zero or both infinite, differentiate numberator and denominator and take the limit of the new function. Repeat as many times as needed as long as it satisfies the conditions. Note that $x_0$ may be replaced by $\pm\infty$ or $x_0\pm$

Theorem 26. Cauchy Mean Value Theorem

Let f,g be continuous on $[a,b]$ and differentiable on $(a,b)$ with $g(a)\neq g(b)$. Then there exists $c$ in $(a,b)$ such that $g’(c)\frac{f(b)-f(a)}{g(b)-g(a)}=f’(c)$

Integration

Anti-derivative and Geometrical Interpretation

Definition 11.

Given $f(x)$ defined over some interval then if we can find a function $F(x)$ defined over the same interval such that

$F’(x)=f(x)$ then $F(x)$ is the indefinite integral of $f$ $\longrightarrow F = \int f(x)dx$. Then

$\frac{d}{dx}(F-G)=0\Rightarrow F(x) = G(x)+ constant$

Area under a curve

Theorem 27.

The function $F(x)$ is differentiable and its derivatives is equal to $f(x)$. Another way to state this is $\frac{d}{dx}\int^x_af(t)dt=f(x)$

Definition 12. Signed Area

If $f(x)<0$ then the area is below the $x-axis$.Define $F(x)$ to be minus the area. This leads to the definite integral.

The Riemann Sum

Given $f(x),a\leq x\leq b$, take the partition of the interval $[a.b]$ to be $x_i = a+ih$ $i=0,1,\ldots\ldots,n$ $h=\frac{b-a}{n}$ Take any sub-interval $[x_{i-1},x_i]$ and let $x_i \in [x_{i-1},x_i]$. Then the Riemann sum is $\Sigma^n_{i=1} f(x_i)h$ There are three ways of picking $x_i$

  1. $x_i*=x_i$ the right hand Riemann Summ

  2. $x_i*=x_{i-1}$ left hand RS

  3. $x_i*=\frac{1}{2}(x_i+x_{i-1})$ mid point RS

The Limit as $n\to \infty,h\to0$ $\lim_{n\to \infty}\sum^n_{i=1}f(x_i*)h=\int^b_af(x)dx$ This can be probed using squeeze theorem between the Lower Riemann sum and Right Sum

Properties of the definite Integral; Fundamental Theorrem of Calculus

  1. $\int^b_acf(x)dx =c\int^b_af(x)dx$ $c$ a constant

  2. $\int^b_af(x)+g(x)dx = \int^b_af(x)dx+\int^b_ag(x)dx$

  3. If $c\in (a,b)$ then

    $\int^b_af(x)dx=\int^c_af(x)dx+\int^b_cf(x)dx$

  4. If $f(x)\leq g(x)$ for $x\in [a,b]$ then

    $\int^b_af(x)dx\leq \int^b_ag(x)dx$

Theorem 28. Suppose $g(x)$ is defined for all $x\in [a,b]$ and is differentiable on $[a,b]$. Then

$\int^b_ag’(x)dx = g(b)-g(a)$

\

Theorem 29. Fundamental Theorem of Calculus

Suppose $F$ is differentiable on [a.b] and $F’$ is integrable on $[a,b]$ Then

$\int^b_aF’(x)dx= F(b)-F(a)$

If $f$ is integrable on $[a,b]$ and has anti-derivative $F$ then

$\int^b_af(x)dx=F(b)-F(a)$ Useful Theorem $\frac{d}{dx}\int^{g(x)}_af(t)dt=f(g(x))g’(x)$

Some Application

just do practise questions for these. this aint fucking physics note

Improper Integrals

Definition 13. Improper Integral

$\int^b_af(x)dx$ is an improper integral if
(i) $a=-\infty$ and/or $b=\infty$
(ii) $f(x)\to \pm\infty \text{ in } (a,b)$

Mean value theorem for Integrals

Given a function $f$ that is integrable on $[a.b]$ we define its average $\langle f \rangle_{[a,b]}$ by the formula $\langle f \rangle_{[a,b]} = \frac{1}{b-a}\int^b_af(x)dx$\

Theorem 30.

Let $f$ be continuous on $[a,b]$ then there exists a point $x_0\in(a.b)$ such that

center $f(x_0)=\frac{1}{b-a}\int^b_af(x)dx$

Techniques of Integration

lmao it’s just integration just get good

Application of Integration

Length of Curves $L = \int^b_a[1+(f’(x))^2]^{1/2}dx$
$L=\int^{t_1}_{t_0}[(\frac{dx}{dt})^2+(\frac{dy}{dt})^2]^{1/2}dt$ Volumes of Revolution

$V=\int^b_a\pi(f(x))^2dx$ Rotating around x-axis
$V=\int^b_a2\pi xf(x)dx$ Revolving around the y-axis
swith the x and y for rotating around y-axis

Surface area of revolution $S=\int^b_a2\pi f(x)\sqrt{1+(f(x))^2}dx$

Centre of Mass

1D case-simple If centre of mass is at x=|x, then we must have zero total moment

ecnter $\Sigma m_k(\bar{x}-x_k)=0$ i.e $\bar{x}=\frac{\Sigma m_kx_k}{\Sigma m_k}$

2D-case- disrete masses
If there are n-masses of mass $m_k$ and cordinates $(x_k,y_k)$. Assume the centre of mass is $(\bar(x),\bar{y})$. So must balance the moments in $x-axis$ and $y-axis$
Therefore:
$\bar{x} = \frac{\Sigma m_ix_i}{\Sigma m_i}$,$\bar{y} = \frac{\Sigma m_iy_i}{\Sigma m_i}$
Now for continuous mass distribution.
Define the density per unit area as $\rho(x,y)$
Dividing the region into small rectangles with sides $\Delta y, \Delta x$
So the moment of one of these rectangles about the y-axis is $x_i\rho(x_i,y_i)\Deltax \Delta y$
Adding all of them gives $\Sum_i\Sum_jx_i\rho(x_i,y_i)\Deltax \Delta y$
The moment of the whole plate about the y-axis $\bar{x\int\int\rho(x,y)dxdy}$
Therefore $\bar{x}\int\int\rho dxdy = \int\int x \rho dxdy$
similar result for $\bar{y}$

\

Theorem 31. Theorem of Pappus

Let $R$ be a region that lies on one side of line $l$ $A=\text{area of }R$
$V=$ Volume obtained by rotating about $l$
$d=$ distane travelled by the centre of mass when $R$ is rotated
then $V=Ad$

Mment of Inertia

Consider an object of mass $m$ at a distane $y$ from the x-axis rotating at an angular velocity of $\omega$.
Then the velocity of the the object is
$v=y\omega$
And thus the kinetic energy of the object is
$KE=\frac{1}{2}m(y\omega)^2$.
The coefficient of $\frac{1}{2}\omega^2$ ia defined to be the moment of inertia. Hence for the single particle considered here, we define the moment of inertia $I$ to be
$I=my^2$
And therefore $KE=\frac{1}{2}\omega^2I$

So using this, we an express the moment of inertia of a string.
Moment of Inertia about $x$-axis - $I_x = \int^{x_1}{x_0}\rho (x)y^2\sqrt{1+y^2}$
Moment of Inertia about $y$-axis - $I_y = \int^{x_1}{x_0}\rho (x)x^2\sqrt{1+y^2}$

Length Of curves and areas ousing polar coorindates

Length of polar curve: $L=\int^{\beta}{\theta =\alpha}[(\frac{dr}{d\theta})^2+r^2]^{1/2}d\theta$
Area of polar curve
$A=\frac{1}{2}\int^{\beta}{alpha} r^2 d\theta$

Series

Definition 14.

Given a sequence ${a_n}{n\geq1}$of real numbers, define the sequence of partial sums $S_N=\Sigma^N{n=1}a_n$ If $S_N\to S$ as $N\to\infty$ we say the series converges to the sum $S$ $S=\Sigma^{\infty}_{n=1}a_n$

Theorem 32.

The series $\sum^{\infty}_{n=1}\frac{1}{n}$ diverges to $+\infty$

Theorem 33.

If $\alpha >1$ is a rational number, then $\sum^{\infty}_{n=1}\frac{1}{n^{\alpha}}$ converges

Elemental algebraic rules for series

Theorem 34.

If the series $\sum^{\infty}_{n=1}a_n$ converges then $a_n \to 0$ as $n \to \infty$

Cauchy sequences and convergence of series

Definition 15. Cauchy Sequence

Cauchy Sequence if and only if:

$\forall \epsilon>0 \exist N\in \mathbb{N}\text{ such that for any } m,n>N$
$|S_m-S_n|<\epsilon$

Theorem 35. Every cauchy sequence converges

\

Theorem 36. The alternating series test

A series thats alternating and $a_n\to 0$ as $n\to \infty$ converges

Convergence tests

Theorem 37. Comparison test

Llet $\sum^{\infty}_{n=1}b_n$ be convergent with $b_n$ non-negative. If $|a_n|\leq b_n$then the series for $a_n$ converges

Theorem 38.

Every absolutely convergent series is convergent

Theorem 39. Integral test

Let $f(x)$ be a function which is defined for all $x\geq1$, and is positive and decreasing. $\sum^{\infty}_{n=1}f(n)$ converges if and only if the indefinite integral to infinity converges

Theorem 40. The ratio test

Let $S= \sum^{\infty}{n=1}a_n$ $\lim{n\to \infty}|\frac{a{n+1}}{a_n}|=L$ Then:

  1. If $L<1$ the series converges absolutely

  2. If $L>1$ the series diverges

  3. If $L=1$ the test is inconclusive

Theorem 41. The root test

Suppose:

center $\lim{n\to \infty}|a_n|^{1/n}=L$

Then:\

  1. If $L<1$ the series converges absolutely

  2. If $L>1$ the series diverges

  3. If $L=1$ the test is inconclusive

Power Series

Definition 16.

let $x$ be a real number and ${a_n}{n\geq0}$ be a sequence of numbers. Then we can form the power series $\sum^{\infty}{n=0}a_nx^n$. The partial sums $S_N$ are polynomials of degree $N$\

Theorem 42.

Assume that there is a number $R>0$ such that $\sum^{\infty}{n=0}a_nR^n$ converges. Then for all $|x|<R$ the series$\Sum^{\infty}{n=0}a_nx^n$ converges absolutely\

Definition 17.

The greatest such $R$(mentioned above) is called the radius of convergence.

Theorem 43. Ratio test for power series

Let $\Sum^{\infty}{n=0}a_n$ be a power series and asuume that $\lim{n\to\infty}|\frac{a{n+1}}{a_n}|=L$ exists. Let $R=\frac{1}{L}$
Then

  1. If $x<R$ the series converges absolutely
  2. If $x>R$ the serues diverges
  3. If $x=\pm R$ could converge or diverge

\

Differentiation and integration of power series

We can differentiate power series if $x<R$

Theorem 44.

Let $f(x)=\sum^{\infty}{n=0}a_n$
THen $f’(x)=\sum^{\infty}{n=0}na_{n-1}$ The integral is the opposite

Theorem 45.

If two power series with radi of convergence $R_1,R_2$ are added or multiplied together then the radi of convergence of the new series is $min(R_1,R_2)$

Taylor series

Theorem 46. Taylor’s theorem with remainder

Let $f$ be a function defined on a closed interval between two numbers $x_0$ and $x$. Assume that the function has $n+1$ derivatigves on the

center $f(x) = f(x_0)+f’(x_0)(x-x_0)+\frac{f^{2}(x_0)}{2!}(x-x_0)+\ldots +\frac{f^{n}(x_0)}{n!}(x-x_0)^n+R_n$

where the remaineder $R_n$ is given by

center $R_n= \int^x_{x_0}\frac{(x-t)^n}{n!}f^{n+1}(t)dt$

Exponentials and logarithms. Binomial theorem

$e^x=1+x+\frac{x^2}{2}+\ldots+\frac{x^n}{n!}+\frac{e^c}{(n+1)!}x^{n+1}$
$log(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}+\ldots+(-1)^{n-1}\frac{x^n}{n}+(-1)^{n}\int^x_0\frac{t^n}{1+t}dt$
$(1+x)^n=\sum^{\infty}_{n=1}\frac{\alpha(\alpha-1)\ldots(\alpha -n+1)}{n!}x^n$

Orthogonal and orthonormal function spaces

Definition 18.

If $f,g$ are real value functions that are Riemann integrabale then the inner product of $f,g$ are

center $(f,g)=\int^b_af(x)g(x)dx$

\

Definition 19.

Let $\mathcal{S}={\phi_0,\phi_1,\ldots,}$ be a collection of functions that are Riemann integrable on [a,b] If

center $(\phi_n,\phi_m)=0$ whenver $m\neq n$

Then $\mathcal{S}$ is an Orthogonal system on [a,b]. Additionally if $||\phi_n||=1$ then $\mathcal{S}$ is said to be Orthonormal

Periodic functiuons and periodic extensions

dfn

At points of discontinuity define

center $f(\xi) = \frac{1}{2}[f(\xi +)+f(\xi-)]$

\

Trigonometric polynomials

Euler’s relation

$cos(\theta)+isin(\theta)=e^{i\theta}$ Orthogonality $\int^{\pi}_{-\pi}e^{inx}e^{-imx}dx= 0$ if $n\neqm, =2\pi$if $n=m$

Fourier series

Consider the trigonetrix polynomial

center $f(x)=S_N(x)= \frac{1}{2}a_0+\Sum^N_{n=1}a_ncos(nx)+b_nsin(nx)$
where $a_n = \frac{1}{\pi}\int^{\pi}{-\pi}f(x)cos(nx)dx$
$b_n = \frac{1}{\pi}\int^{\pi}{-\pi}f(x)sin(nx)dx$

Orthogonality properties If $m,n$ are integers then

center $\int^{\pi}{-\pi}sin(mx)sin(nx)dx =\int^{\pi}{-\pi}cos(mx)cos(nx)dx=0,m\neq n,\pi, m=n$
$\int^{\pi}_{-\pi}sin(mx)cos(nx)dx=0$

Theorem 47. The fourier series formed by the fourier coefficients converges to the value $f(x)$ for any piecewise continuous function $f(x)$ over period period $2\pi$ which has piecewise continuous derivatives of first and second order. At any discontinuities the function must be defined by $f(x) = \frac{1}{2}[f(x^+)f(x^-)]$


If $c_n(x)=\frac{1}{2}+ cos(x)+cos(2X)+cos(3x)+\ldots = \frac{sin(n+\frac{1}{2})x}{2sin(0.5x)}$ Define the poits where $\frac{1}{2}x = n\pi$ define $c_n$ by $n+1/2$ Riemann-Lebesgue Lemma

$I_{\lambda}= \int^b_ag(x)sin(\lambda x)dx$ tends to 0 as $\lamda \to \infty$

Lemma 2

$\int^{\infty}_0\frac{sin(z)}{z}dz=\frac{\pi}{2}$

Parseval’s indentity
If $f(x)=S_N(x)= \frac{1}{2}a_0+\sum^N_{n=1}a_ncos(nx)+b_nsin(nx)$
Then $\frac{1}{\pi}\int^{\pi}{-pi}f^2dx=\frac{1}{2}a^2+0+\sum^{\infty}{n=1}(a_n^2+b_n^2)$

Fourier Transform pair

$f(x)=\frac{1}{2\pi}\int^{\infty}{-\infty}\int^{\infty}{-\infty}f(t)e^{i\omega t}dte^{i\omega x d\omega}$