The document discusses Chebyshev polynomials. It defines Chebyshev polynomials as orthogonal polynomials related to de Moivre's formula that can be defined recursively. It provides key properties of Chebyshev polynomials including that they are the extremal polynomials for many properties and important in approximation theory. The document also provides formulas for generating Chebyshev polynomials, their orthogonality properties, and their use in representing functions through orthogonal series expansions.
Mathematics and History of Complex VariablesSolo Hermelin
Mathematics of complex variables, plus history.
This presentation is at a Undergraduate in Science (Math, Physics, Engineering) level.
Please send comments and suggestions to solo.hermelin@gmail.com, thanks! For more presentations, please visit my website at http://www.solohermelin.com
This presentation will be very helpful to learn about system of linear equations, and solving the system.It includes common terms related with the lesson and using of Cramer's rule.
Please download the PPT first and then navigate through slide with mouse clicks.
Mathematics and History of Complex VariablesSolo Hermelin
Mathematics of complex variables, plus history.
This presentation is at a Undergraduate in Science (Math, Physics, Engineering) level.
Please send comments and suggestions to solo.hermelin@gmail.com, thanks! For more presentations, please visit my website at http://www.solohermelin.com
This presentation will be very helpful to learn about system of linear equations, and solving the system.It includes common terms related with the lesson and using of Cramer's rule.
Please download the PPT first and then navigate through slide with mouse clicks.
In this presentation we will learn Del operator, Gradient of scalar function , Directional Derivative, Divergence of vector function, Curl of a vector function and after that solved some example related to above.
Gradient in math
Directional derivative in math
Divergence in math
Curl in math
Gradient , Directional Derivative , Divergence , Curl in mathematics
Gradient , Directional Derivative , Divergence , Curl in math
Gradient , Directional Derivative , Divergence , Curl
In this lecture, we will discuss:
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
ppt on Vector spaces (VCLA) by dhrumil patel and harshid panchalharshid panchal
this is the ppt on vector spaces of linear algebra and vector calculus (VCLA)
contents :
Real Vector Spaces
Sub Spaces
Linear combination
Linear independence
Span Of Set Of Vectors
Basis
Dimension
Row Space, Column Space, Null Space
Rank And Nullity
Coordinate and change of basis
this is made by dhrumil patel which is in chemical branch in ld college of engineering (2014-18)
i think he is the best ppt maker,dhrumil patel,harshid panchal
What is Fourier Transform
Spatial to Frequency Domain
Fourier Transform
Forward Fourier and Inverse Fourier transforms
Properties of Fourier Transforms
Fourier Transformation in Image processing
For a system involving two variables (x and y), each linear equation determines a line on the xy-plane. Because a solution to a linear system must satisfy all of the equations, the solution set is the intersection of these lines, and is hence either a line, a single point, or the empty set
In this presentation we will learn Del operator, Gradient of scalar function , Directional Derivative, Divergence of vector function, Curl of a vector function and after that solved some example related to above.
Gradient in math
Directional derivative in math
Divergence in math
Curl in math
Gradient , Directional Derivative , Divergence , Curl in mathematics
Gradient , Directional Derivative , Divergence , Curl in math
Gradient , Directional Derivative , Divergence , Curl
In this lecture, we will discuss:
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
ppt on Vector spaces (VCLA) by dhrumil patel and harshid panchalharshid panchal
this is the ppt on vector spaces of linear algebra and vector calculus (VCLA)
contents :
Real Vector Spaces
Sub Spaces
Linear combination
Linear independence
Span Of Set Of Vectors
Basis
Dimension
Row Space, Column Space, Null Space
Rank And Nullity
Coordinate and change of basis
this is made by dhrumil patel which is in chemical branch in ld college of engineering (2014-18)
i think he is the best ppt maker,dhrumil patel,harshid panchal
What is Fourier Transform
Spatial to Frequency Domain
Fourier Transform
Forward Fourier and Inverse Fourier transforms
Properties of Fourier Transforms
Fourier Transformation in Image processing
For a system involving two variables (x and y), each linear equation determines a line on the xy-plane. Because a solution to a linear system must satisfy all of the equations, the solution set is the intersection of these lines, and is hence either a line, a single point, or the empty set
All the best to all students of class IX...This PPT will makes your difficulties easy to do....You will understand the polynomial chapter easily by seeing this ....Thanks for watching this ..Please Share, Like and Subscribe the PPT
Introduction, History
Application in different fields like CAD, schematic capture, medicine, art etc.
Hardware concepts
2-D and 3-D algorithms
Line drawing, viewing transformations, other transforms (scaling, rotation, translation)
Curve Modeling
Mathematics of generation of parametric cureve like Bazier, Spline, Hermite
3D Object Modeling
Visible Surface Detection and Surface Rendering
Introduction to Animation
Text Book
Computer Graphics C Version – Hearn & Baker
A brief and easy concept of Simple harmonic oscillator. How we can get simple harmonic motion equation from Lagrange's equation of motion. How can we obtain this from Lagrange's equation of motion.
Lecture 8: Introduction to Quantum Chemical Simulation graduate course taught at MIT in Fall 2014 by Heather Kulik. This course covers: wavefunction theory, density functional theory, force fields and molecular dynamics and sampling.
I am Frank P. I am a Physical Chemistry Assignment Expert at eduassignmenthelp.com. I hold a Ph.D. in Physical Chemistry, from Malacca, Malaysia. I have been helping students with their homework for the past 6 years. I solve assignments related to Physical Chemistry.
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A Tau Approach for Solving Fractional Diffusion Equations using Legendre-Cheb...iosrjce
In this paper, a modified numerical algorithm for solving the fractional diffusion equation is
proposed. Based on Tau idea where the shifted Legendre polynomials in time and the shifted Chebyshev
polynomials in space are utilized respectively.
The problem is reduced to the solution of a system of linear algebraic equations. From the computational point
of view, the solution obtained by this approach is tested and the efficiency of the proposed method is confirmed.
Generating a high quality Chaotic sequence is crucial to the success of the Superefficient Monte Carlo Simulation methodology. In this slides, we discuss how to numerically generates Chebychev Chaotic Sequence with arbitrary precision, and proposed a highly efficient parallel implementation.
I am George P. I am a Chemistry Assignment Expert at eduassignmenthelp.com. I hold a Ph.D. in Chemistry, from Perth, Australia. I have been helping students with their homework for the past 6 years. I solve assignments related to Chemistry.
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On Application of Power Series Solution of Bessel Problems to the Problems of...BRNSS Publication Hub
One of the most powerful techniques available for studying functions defined by differential equations is to produce power series expansions of their solutions when such expansions exist. This is the technique I now investigated, in particular, its feasibility in the solution of an engineering problem known as the problem of strut of variable moment of inertia. In this work, I explored the basic theory of the Bessel’s function and its power series solution. Then, a model of the problem of strut of variable moment of inertia was developed into a differential equation of the Bessel’s form, and finally, the Bessel’s equation so formed was solved and result obtained.
Lecture13p.pdf.pdfThedeepness of freedom are threevalues.docxcroysierkathey
Lecture13p.pdf.pdf
Thedeepness of freedom are threevalues at thenude
functional Notconforming
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data Yifp o is the Zeno poly
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LECTURE # 3:
ABSTRACT RITZ-GALERKIN METHOD
MATH610: NUMERICAL METHODS FOR PDES:
RAYTCHO LAZAROV
1. Variational Formulation
In the previous lecture we have introduced the following space of functions
defined on (0, 1):
(1)
V =
v :
v(x) is continuous function on (0, 1);
v′(x) exists in generalized sense and in L2(0, 1);
v(0) = v(1) = 0
:= H10 (0, 1)
and equipped it with the L2 and H1 norms
‖v‖ = (v,v)1/2 and ‖v‖V = (v,v)
1/2
V =
(∫ 1
0
(u′2 + u2)dx
)1
2
.
We also introduced the following variational and minimization problems:
(V ) find u ∈ V such that a(u,v) = L(v), ∀ v ∈ V,
(M) find u ∈ V such that F(u) ≤ F(v), ∀ v ∈ V,
where a(u,v) is a bilinear form that is symmetric, coercive and contin-
uous on V and L(v) is continuous on V and F(v) = 1
2
a(u,u) −L(v).
As an example we can take
a(u,v) ≡
∫ 1
0
(k(x)u′v′ + q(x)uv) dx and L(v) ≡
∫ 1
0
f(x)v dx.
Here we have assumed that there are positive constants k0, k1, M such that
(2) k1 ≥ k(x) ≥ k0 > 0, M ≥ q(x) ≥ 0, f ∈ L2(0, 1).
These are sufficient for the symmetry, coercivity and continuity of the
bilinear form a(., .) and the continuity of the linear form L(v).
The proof of these properties follows from the following theorem:
Theorem 1. Let u ∈ V ≡ H10 (0, 1). Then the following inequalities are
valid:
(3)
|u(x)|2 ≤ C1
∫ 1
0
(u′(x))2dx for any x ∈ (0, 1),∫ 1
0
u2(x)dx ≤ C0
∫ 1
0
(u′(x))2dx.
with constants C0 and C1 that are independent of u.
1
2 MATH610: NUMERICAL METHODS FOR PDES: RAYTCHO LAZAROV
Proof: We give two proofs. The simple one proves the above inequali-
ties with C0 = 1/2 and C1 = 1. The better proof establishes the above
inequalities with C0 = 1/6 and C1 = 1/4.
Indeed, for any x ∈ (0, 1) we have:
u(x) = u(0) +
∫ x
0
u′(s)ds.
Since u ∈ H10 (0, 1) then u(0) = 0. We square this equality and apply
Cauchy-Swartz inequality:
(4) |u(x)|2 =
∣∣∣∫ x
0
u′(s)ds
∣∣∣2 ≤ ∫ x
0
1ds
∫ x
0
(u′(s))2ds ≤ x
∫ x
0
(u′(s))2ds.
Taking the maximal value of x on the right hand side of this inequality
w ...
Lecture13p.pdf.pdfThedeepness of freedom are threevalues.docxjeremylockett77
Lecture13p.pdf.pdf
Thedeepness of freedom are threevalues at thenude
functional Notconforming
patrtaf.us
vi sci x
I beease
ittouch
41 u VCsci inhalfedgeL
U VCI't x
Since u CPz are sci sc 7 that it have 3 Zeusunless e o
E is P anisolvent it forgiven I lop C P s t 4 p di
same
degree
y i l N Yi C E
Sabi n ofsystem YCp g
This is equivalent to say theonlypolynomial C PthetinterpolateZero
data Yifp o is the Zeno poly
vcpi.POTFF.gg In Edem
e e
I CRIvalue VCR Ca Ya
metfunctor
p
E3
pjJ Chip J
Shun E is p unisolvent
Y Cul VCR7 0
Xz V UCR o
rf VI UCB 0
Then over the edge PP we hone
C P havingtworootsPR
D This implies we 0
If e consider the other to edges G e b thesame
argument we can see Eo tht means W Lv o hersonly trivial
Solution
Then Yi CUI Ri for any Xi
E is P unisowent
y
csiy
Ya
f P Y cnn.PT
III Ldj Pg I Pre 2 ily
a PyO ein a
451214 7 f p i y g d CP f
ftp b f CRI I B so
fickle Cps O y Cp 7 L
Escaple5 in lectureto
Lecture_03_S08.pdf
LECTURE # 3:
ABSTRACT RITZ-GALERKIN METHOD
MATH610: NUMERICAL METHODS FOR PDES:
RAYTCHO LAZAROV
1. Variational Formulation
In the previous lecture we have introduced the following space of functions
defined on (0, 1):
(1)
V =
v :
v(x) is continuous function on (0, 1);
v′(x) exists in generalized sense and in L2(0, 1);
v(0) = v(1) = 0
:= H10 (0, 1)
and equipped it with the L2 and H1 norms
‖v‖ = (v,v)1/2 and ‖v‖V = (v,v)
1/2
V =
(∫ 1
0
(u′2 + u2)dx
)1
2
.
We also introduced the following variational and minimization problems:
(V ) find u ∈ V such that a(u,v) = L(v), ∀ v ∈ V,
(M) find u ∈ V such that F(u) ≤ F(v), ∀ v ∈ V,
where a(u,v) is a bilinear form that is symmetric, coercive and contin-
uous on V and L(v) is continuous on V and F(v) = 1
2
a(u,u) −L(v).
As an example we can take
a(u,v) ≡
∫ 1
0
(k(x)u′v′ + q(x)uv) dx and L(v) ≡
∫ 1
0
f(x)v dx.
Here we have assumed that there are positive constants k0, k1, M such that
(2) k1 ≥ k(x) ≥ k0 > 0, M ≥ q(x) ≥ 0, f ∈ L2(0, 1).
These are sufficient for the symmetry, coercivity and continuity of the
bilinear form a(., .) and the continuity of the linear form L(v).
The proof of these properties follows from the following theorem:
Theorem 1. Let u ∈ V ≡ H10 (0, 1). Then the following inequalities are
valid:
(3)
|u(x)|2 ≤ C1
∫ 1
0
(u′(x))2dx for any x ∈ (0, 1),∫ 1
0
u2(x)dx ≤ C0
∫ 1
0
(u′(x))2dx.
with constants C0 and C1 that are independent of u.
1
2 MATH610: NUMERICAL METHODS FOR PDES: RAYTCHO LAZAROV
Proof: We give two proofs. The simple one proves the above inequali-
ties with C0 = 1/2 and C1 = 1. The better proof establishes the above
inequalities with C0 = 1/6 and C1 = 1/4.
Indeed, for any x ∈ (0, 1) we have:
u(x) = u(0) +
∫ x
0
u′(s)ds.
Since u ∈ H10 (0, 1) then u(0) = 0. We square this equality and apply
Cauchy-Swartz inequality:
(4) |u(x)|2 =
∣∣∣∫ x
0
u′(s)ds
∣∣∣2 ≤ ∫ x
0
1ds
∫ x
0
(u′(s))2ds ≤ x
∫ x
0
(u′(s))2ds.
Taking the maximal value of x on the right hand side of this inequality
w.
I am Grey N. I am a Physical Chemistry Assignment Expert at eduassignmenthelp.com. I hold a Ph.D. in Physical Chemistry, from Calgary, Canada. I have been helping students with their homework for the past 6 years. I solve assignments related to Physical Chemistry.
Visit eduassignmenthelp.com or email info@eduassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Physical Chemistry Assignments.
Overview of the fundamental roles in Hydropower generation and the components involved in wider Electrical Engineering.
This paper presents the design and construction of hydroelectric dams from the hydrologist’s survey of the valley before construction, all aspects and involved disciplines, fluid dynamics, structural engineering, generation and mains frequency regulation to the very transmission of power through the network in the United Kingdom.
Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
(C) 2024 Robbie E. Sayers
Explore the innovative world of trenchless pipe repair with our comprehensive guide, "The Benefits and Techniques of Trenchless Pipe Repair." This document delves into the modern methods of repairing underground pipes without the need for extensive excavation, highlighting the numerous advantages and the latest techniques used in the industry.
Learn about the cost savings, reduced environmental impact, and minimal disruption associated with trenchless technology. Discover detailed explanations of popular techniques such as pipe bursting, cured-in-place pipe (CIPP) lining, and directional drilling. Understand how these methods can be applied to various types of infrastructure, from residential plumbing to large-scale municipal systems.
Ideal for homeowners, contractors, engineers, and anyone interested in modern plumbing solutions, this guide provides valuable insights into why trenchless pipe repair is becoming the preferred choice for pipe rehabilitation. Stay informed about the latest advancements and best practices in the field.
Water scarcity is the lack of fresh water resources to meet the standard water demand. There are two type of water scarcity. One is physical. The other is economic water scarcity.
Sachpazis:Terzaghi Bearing Capacity Estimation in simple terms with Calculati...Dr.Costas Sachpazis
Terzaghi's soil bearing capacity theory, developed by Karl Terzaghi, is a fundamental principle in geotechnical engineering used to determine the bearing capacity of shallow foundations. This theory provides a method to calculate the ultimate bearing capacity of soil, which is the maximum load per unit area that the soil can support without undergoing shear failure. The Calculation HTML Code included.
Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
CFD Simulation of By-pass Flow in a HRSG module by R&R Consult.pptxR&R Consult
CFD analysis is incredibly effective at solving mysteries and improving the performance of complex systems!
Here's a great example: At a large natural gas-fired power plant, where they use waste heat to generate steam and energy, they were puzzled that their boiler wasn't producing as much steam as expected.
R&R and Tetra Engineering Group Inc. were asked to solve the issue with reduced steam production.
An inspection had shown that a significant amount of hot flue gas was bypassing the boiler tubes, where the heat was supposed to be transferred.
R&R Consult conducted a CFD analysis, which revealed that 6.3% of the flue gas was bypassing the boiler tubes without transferring heat. The analysis also showed that the flue gas was instead being directed along the sides of the boiler and between the modules that were supposed to capture the heat. This was the cause of the reduced performance.
Based on our results, Tetra Engineering installed covering plates to reduce the bypass flow. This improved the boiler's performance and increased electricity production.
It is always satisfying when we can help solve complex challenges like this. Do your systems also need a check-up or optimization? Give us a call!
Work done in cooperation with James Malloy and David Moelling from Tetra Engineering.
More examples of our work https://www.r-r-consult.dk/en/cases-en/
About
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Technical Specifications
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
Key Features
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface
• Compatible with MAFI CCR system
• Copatiable with IDM8000 CCR
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
Application
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
3. CHEBYSHEV POLYNOMIALS
In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev,[1] are a sequence
of orthogonal polynomials which are related to de Moivre's formula and which can be defined
recursively. One usually distinguishes between Chebyshev polynomials of the first kind which
are denoted Tn and Chebyshev polynomials of the second kind which are denoted Un. The letter
T is used because of the alternative transliterations of the name Chebyshev as T chebycheff, T
chebyshev (French) or T schebyschow (German).
The Chebyshev polynomials Tn or Un are polynomials of degree n and the sequence of
Chebyshev polynomials of either kind composes a polynomial sequence.
Chebyshev polynomials are polynomials with the largest possible leading coefficient, but
subject to the condition that their absolute value on the interval [-1,1] is bounded by 1. They
are also the extremal polynomials for many other properties.[2]
Chebyshev polynomials are important in approximation theory because the roots of the
Chebyshev polynomials of the first kind, which are also called Chebyshev nodes, are used as
nodes in polynomial interpolation. The resulting interpolation polynomial minimizes the
problem of Runge's phenomenon and provides an approximation that is close to the
polynomial of best approximation to a continuous function under the maximum norm. This
approximation leads directly to the method of Clenshaw–Curtis quadrature.
4. CHEBYSHEV POLYNOMIALS
Differential Equation and Its Solution
The Chebyshev differential equation is written as
(1 − 𝒙 𝟐 )
𝒅 𝟐y
𝒅𝒙 𝟐 − 𝒙
𝒅𝒚
𝒅𝒙
+ 𝒏 𝟐 𝒚 = 𝟎 𝒏 = 𝟎, 𝟏, 𝟐, 𝟑 … … …
If we let x = cos t we obtain
d2 y
dt2
+ 𝒏 𝟐
y = 0
whose general solution is
y = A cos nt + B sin nt
or as
y = A cos(n 𝒄𝒐𝒔−𝟏
x) + B sin(n 𝒄𝒐𝒔−𝟏
x) |x| < 1
or equivalently
y = ATn (x) + BUn (x) |x| < 1
where Tn (x) and Un (x) are defined as Chebyshev polynomials of the
first and second kind of degree n, respectively
5. CHEBYSHEV POLYNOMIALS
If we let x = cosh t we obtain
d2 𝑦
dt2
− 𝑛2
𝑦 = 0
whose general solution is
y = A cosh nt + B sinh nt
or as
y = A cosh(n cosh−1 x) + B sinh(n cosh−1 x) |x| > 1
or equivalently
y = ATn (x) + BUn (x) |x| > 1
The function Tn (x) is a polynomial. For |x| < 1 we have
𝑻 𝒏 (x) + i𝑼 𝒏 (x) = (cos t + i sin t) 𝒏
= ( x +i 𝟏 − 𝒙 𝟐) 𝒏
𝑻 𝒏 (x) − i𝑼 𝒏 (x) = (cos t − i sin t) 𝒏
= ( x − i 𝟏 − 𝒙 𝟐) 𝒏
6. CHEBYSHEV POLYNOMIALS
from which we obtain
𝑻 𝒏 (x)=
𝟏
𝟐
[( x +i 𝟏 − 𝒙 𝟐) 𝒏
+ ( x − i 𝟏 − 𝒙 𝟐) 𝒏
]
For |x| > 1 we have
Tn (x) + Un (x) = 𝒆 𝒏𝒕
= (x± 𝟏 − 𝒙 𝟐) 𝒏
Tn (x) − Un (x) = 𝒆−𝒏𝒕
= (x± 𝟏 − 𝒙 𝟐) 𝒏
The sum of the last two relationships give the same result for Tn (x).
Chebyshev Polynomials of the First Kind of Degree n
The Chebyshev polynomials Tn (x) can be obtained by means of
Rodrigue’s formula
𝑻 𝒏 (x)=
(−𝟐) 𝒏 𝒏!
𝟐𝒏!
𝟏 − 𝒙 𝟐 𝒅 𝒏
𝒅𝒙 𝒏(1-𝒙 𝟐) 𝒏−𝟏/𝟐 n=0,1,2,3,……….
7. CHEBYSHEV POLYNOMIALS
The first twelve Chebyshev polynomials are listed in Table 1 and
then as powers of x in terms of Tn (x) in Table 2.
Table 1: Chebyshev Polynomials of the First Kind
𝑇0 (x)=1
𝑇1 𝑋 = 𝑋
𝑇2 (x)=2𝑥2 − 1
𝑇3 (x)=4𝑥2
− 3𝑥
𝑇4 (x)=8𝑥4 − 8𝑥2 + 1
𝑇5 (x)=16𝑥5
− 20𝑥3
+ 5x
𝑇6 (x)=32𝑥6
− 48𝑥4
+ 18𝑥2
− 1
𝑇7 (x)=64𝑥7 − 112𝑥5 + 18𝑥3 − 7𝑥
𝑇8 (x)=128𝑥8
− 256𝑥6
+ 160𝑥4
− 32𝑥2
+ 1
𝑇9 (x)=256𝑥9
− 576𝑥7
+ 432𝑥5
− 120𝑥3
+ 9𝑥
𝑇10 (x)=512𝑥10
− 1280𝑥8
+ 1120𝑥6
− 400𝑥4
+ 50𝑥2
− 1
𝑇11 (x)=1024𝑥11
− 2816𝑥9
+ 2816 𝑥7
− 1232𝑥5
+ 220𝑥3
− 11𝑥
9. CHEBYSHEV POLYNOMIALS
Generating Function for Tn(x)
The Chebyshev polynomials of the first kind can be developed by means of the
generating function
𝑛=0
∞
𝑇𝑛 𝑥 𝑡 𝑛
=
1−𝑡𝑥
1−2𝑡𝑥+𝑥2
Recurrence Formulas for Tn(x)
When the first two Chebyshev polynomials T0 (x) and T1 (x) are known, all
other polynomials Tn (x), n ≥ 2 can be obtained by means of the recurrence
formula
𝑻 𝒏+𝟏 (x) = 2x𝑻 𝒏 (x) −n 𝑻 𝒏−𝟏(x)
The derivative of Tn (x) with respect to x can be obtained from
(1 − 𝑥2 ) 𝑇𝑛(x)=−𝑛𝑥𝑇𝑛(x)+n 𝑇𝑛−1(x)
10. CHEBYSHEV POLYNOMIALS
Special Values of Tn(x)
The following special values and properties of Tn (x) are often useful:
𝑇𝑛 (−x) = (−1 ) 𝑛
𝑇𝑛 (x) 𝑇2𝑛 (0) = (−1 ) 𝑛
𝑇𝑛 (1) = 1 𝑇2𝑛+1 (0) = 0
𝑇𝑛 (−1) = (−1) 𝑛
Orthogonality Property of 𝑇𝑛 (x)
We can determine the orthogonality properties for the Chebyshev
polynomials of the first kind from our knowledge of the orthogonality of
the cosine functions, namely,
0
𝜋
cos(mθ) cos(n θ) dθ =
0 (m ≠ n)
𝜋
2 (m = n ≠ 0)
𝜋 (m = n = 0)
11. CHEBYSHEV POLYNOMIALS
Then substituting
𝑇𝑛 (x) = cos(nθ)
cos θ = x
to obtain the orthogonality properties of the Chebyshev polynomials:
−1
1 𝑇 𝑚 𝑥 𝑇 𝑛 𝑥 𝑑𝑥
1−𝑥2
=
0 (m ≠ n)
𝜋
2 (m = n ≠ 0)
𝜋 (m = n = 0)
We observe that the Chebyshev polynomials form an orthogonal set on
the interval −1 ≤ x ≤ 1 with the weighting function (1 − 𝑥2)
−1
2
12. CHEBYSHEV POLYNOMIALS
Orthogonal Series of Chebyshev Polynomials
An arbitrary function f (x) which is continuous and single-valued, defined over the interval
−1 ≤ x ≤ 1, can be expanded as a series of Chebyshev polynomials
f (x) = 𝐴0 𝑇0(x) + 𝐴1 𝑇1 (x) + 𝐴2 𝑇2 (x) + . . .
= 𝑛=0
∞
𝐴 𝑛 𝑇𝑛 (x)
where the coefficients An are given by
𝐴0=
1
𝜋 −1
1 𝑓 𝑥 𝑑𝑥
1−𝑥2
𝑛 = 0
And
𝐴0=
2
𝜋 −1
1 𝑓 𝑥 𝑇𝑛(𝑥)𝑑𝑥
1−𝑥2
𝑛 = 1,2,3, … .
13. CHEBYSHEV POLYNOMIALS
The following definite integrals are often useful in the series expansion
of f (x):
−1
1 𝑑𝑥
1−𝑥2
= 𝜋 −1
1 𝑥3 𝑑𝑥
1−𝑥2
= 0
−1
1 𝑥𝑑𝑥
1−𝑥2
= 0 −1
1 𝑥4 𝑑𝑥
1−𝑥2
=
3𝜋
8
−1
1 𝑥2 𝑑𝑥
1−𝑥2
=
𝜋
2 −1
1 𝑥5 𝑑𝑥
1−𝑥2
= 0
Chebyshev Polynomials Over a Discrete Set of Points
A continuous function over a continuous interval is often replaced by a set of
discrete values of the function at discrete points. It can be shown that the Chebyshev
polynomials Tn (x) are orthogonal over the following discrete set of N + 1 points xi,
equally spaced on θ,
𝜃𝑖=0,
𝜋
𝑁
,
2𝜋
𝑁
……(N−1)
𝜋
𝑁
,𝜋 where xi = arccos θi
14. CHEBYSHEV POLYNOMIALS
We have
The Tm(x) are also orthogonal over the following N points ti equally spaced
𝜃𝑖=0,
𝜋
2𝑁
,
3𝜋
2𝑁
.
5𝜋
2𝑁
,
(𝑁−1)𝜋
2𝑁
𝑡𝑖= arccos𝜃𝑖
𝑖=1
𝑁
𝑇 𝑚(𝑡𝑖)𝑇𝑛 𝑡𝑖 𝑑𝑥=
0 (𝑚 ≠ 𝑛)
𝑁
2 (𝑚 = 𝑛 ≠ 0)
𝑁 (𝑚 = 𝑛 = 0)
15. CHEBYSHEV POLYNOMIALS
The set of points ti are clearly the midpoints in θ of the first case. The unequal
spacing of the points in xi(N ti) compensates for the weight factor
W(x)= (1 − 𝑥2
)
−1
2
in the continuous case.
Additional Identities of Chebyshev Polynomials
The Chebyshev polynomials are both orthogonal polynomials and the trigonometric
cos nx functions in disguise, therefore they satisfy a large number of useful
relationships.
The differentiation and integration properties are very important in analytical and
numerical work. We begin with
𝑇𝑛+1 (x) = cos[(n + 1) 𝑐𝑜𝑠−1
x]
and
𝑇𝑛−1 (x) = cos[(n - 1) 𝑐𝑜𝑠−1 x]
17. CHEBYSHEV POLYNOMIALS
Therefore
𝑇2 (x)=4𝑇1
𝑇1 (x)=𝑇0
𝑇0 (x)=0
We have the formulas for the differentiation of Chebyshev polynomials,
therefore these formulas can be used to develop integration for the Chebyshev
polynomials
18. CHEBYSHEV POLYNOMIALS
The Shifted Chebyshev Polynomials
For analytical and numerical work it is often convenient to use the half interval 0 ≤ x ≤ 1
instead of the full interval −1 ≤ x ≤ 1. For this purpose the shifted Chebyshev polynomials
are defined:
𝑻
∗
𝒏
(x)=𝑻 𝒏 *(2x-1)
Thus we have for the first few polynomials
𝑻
∗
𝟎
(x)=1
𝑻
∗
𝟏
𝑋 = 2𝑋 −1
𝑻
∗
𝟐
(x)=8𝑥2 − 8𝑥 + 1
𝑻
∗
𝟑
(x)=32𝑥3 − 48𝑥2 + 18𝑥 − 1
𝑻
∗
𝟒
(x)=128𝑥4
− 256𝑥3
+ 160𝑥2
− 32𝑥 + 1
19. CHEBYSHEV POLYNOMIALS
and the following powers of x as functions of 𝑻
∗
𝟎
(x);
1=𝑻
∗
𝟎
x=
1
2
(𝑻
∗
𝟎
+ 𝑻
∗
𝟎
)
𝑥2
=
1
8
(3𝑻
∗
𝟎
+ 𝟒𝑻
∗
𝟏
+𝑻
∗
𝟐
)
𝑥3
=
1
32
(10𝑻
∗
𝟎
+ 1𝟓𝑻
∗
𝟏
+𝟔𝑻
∗
𝟐
+𝑻
∗
𝟑
)
𝑥4=
1
128
(35𝑻
∗
𝟏
+ 56𝑻
∗
𝟐
+28𝑻
∗
𝟑
+𝟖𝑻
∗
𝟒
+𝑻
∗
𝟒
)
The recurrence relationship for the shifted polynomials is:
Or
where
20. CHEBYSHEV POLYNOMIALS
Expansion of 𝑥 𝑛
in a Series of 𝑇𝑛 (x)
A method of expanding 𝑥 𝑛
in a series of Chebyshev polynomials employes the
recurrence relation written as
x 𝑇0 (x) = 𝑇1 (x)
To illustrate the method, consider 𝑥4
This result is consistent with the expansion of x4 given in Table 2
21. CHEBYSHEV POLYNOMIALS
Approximation of Functions by Chebyshev Polynomials
Sometimes when a function f (x) is to be approximated by a polynomial of the form
where |𝑬 𝒏 (x)| does not exceed an allowed limit, it is possible to reduce the degree of the
polynomial by a process called economization of power series. The procedure is to
convert the polynomial to a linear combination of Chebyshev polynomials:
It may be possible to drop some of the last terms without permitting the error to
exceed the prescribed limit. Since |Tn (x)| ≤ 1, the number of terms which can be
omitted is determined by the magnitude of the coefficient b.
22. CHEBYSHEV POLYNOMIALS
The Chebyshev polynomials are useful in numerical work for the interval −1
≤ x ≤ 1 because
1. |Tn (x)] ≤ 1 within −1 ≤ x ≤ 1
2. The maxima and minima are of comparable value
3. The maxima and minima are spread reasonably uniformly over the interval −1
≤ x ≤ 1
4. All Chebyshev polynomials satisfy a three term recurrence relation
5. They are easy to compute and to convert to and from a power series form.
These properties together produce an approximating polynomial which
minimizes error in its application. This is different from the least squares
approximation where the sum of the squares of the errors is minimized; the
maximum error itself can be quite large. In the Chebyshev approximation,
the average error can be large but the maximum error is minimized.
Chebyshev approximations of a function are sometimes said to be mini-max
approximations of the function.
23. CHEBYSHEV POLYNOMIALS
The following table gives the Chebyshev polynomial approximation of several
power series.
Table 3: Power Series and its Chebyshev Approximation
26. CHEBYSHEV POLYNOMIALS
For easy reference the formulas for economization of power series in
terms of Chebyshev are given in Table 4.
Assigned Problems
Problem Set for Chebyshev Polynomials
1-Obtain the first three Chebyshev polynomials 𝑇0 (x), 𝑇1 (x) and 𝑇2 (x)
by means of the Rodrigue’s formula
2. Show that the Chebyshev polynomial 𝑇3 (x) is a solution of
Chebyshev’s equation of order 3.
3. By means of the recurrence formula obtain Chebyshev polynomials
𝑇2 (x) and 𝑇3 (x) given 𝑇0 (x) and 𝑇1 (x).
4. Show that 𝑇𝑛(1) = 1 and 𝑇𝑛 (−1) = (−1) 𝑛
5. Show that 𝑇𝑛 (0) = 0 if n is odd and (−1)
𝑛
2 if n is even.
6. Setting x = cos θ show that
27. CHEBYSHEV POLYNOMIALS
Where i= −1
7. Find the general solution of Chebyshev’s equation for n = 0
8. Obtain a series expansion for f (x) = 𝑥2
in terms of Chebyshev
polynomials 𝑇𝑛 (x),
9. Express x4 as a sum of Chebyshev polynomials of the first kind
28. HERMITE POLYNOMIALS
In mathematics, the Hermite polynomials are a classical orthogonal polynomial
sequence that arise in probability, such as the Edgeworth series; in combinatorics, as an
example of an Appell sequence, obeying the umbral calculus; and in physics, where
they give rise to the eigenstates of the quantum harmonic oscillator. They are named in
honor of Charles Hermite.
Definition
The Hermite polynomials are defined either by
(the "probabilists' Hermite polynomials"), or sometimes by
29. HERMITE POLYNOMIALS
(the "physicists' Hermite polynomials"). These two definitions are not exactly equivalent; either is a
rescaling of the other, to wit
These are Hermite polynomial sequences of different variances; see the material on variances below.
Below, we usually follow the first convention. That convention is often preferred by probabilists
because
1
2𝜋
𝑒−𝑥2/2
is the probability density function for the normal distribution with expected value 0 and
standard deviation
1.
34. HERMITE POLYNOMIALS
Properties
Hn is a polynomial of degree n. The probabilists' version has leading coefficient
1, while the physicists' version has leading coefficient 2n.
Orthogonality
Hn(x) is an nth-degree polynomial for n = 0, 1, 2, 3, .... These polynomials are respect to
Orthogonality with the weight function (measuer)
W(x)= 𝒆−𝒙 𝟐/𝟐
(probabilist)
or
W(x)= 𝒆−𝒙 𝟐
(physicist)
i.e., we have
−∞
∞
𝐻 𝑀(𝑥)𝐻 𝑁(x) 𝒆−𝒙 𝟐/𝟐
dx=n! 2𝜋 (probabilist)
35. HERMITE POLYNOMIALS
OR
−∞
∞
𝐻 𝑀(𝑥)𝐻 𝑁(x) 𝒆−𝒙 𝟐/𝟐
dx=n! 𝜋 (physicist)
The probabilist polynomials are thus orthogonal with respect to the standard
normal probability density function.
Completeness
The Hermite polynomials (probabilist or physicist) form an orthogonal basis of the Hilbert
space of functions satisfying
−∞
∞
|𝒇(𝒙)| 𝟐
w(x) dx< ∞
in which the inner product is given by the integral including the Gaussian weight function
w(x) defined in the preceding section,
𝑓(𝑔) = −∞
∞
𝑓 𝑥 𝑤 𝑥 𝑔 𝑥 𝑑𝑥
An orthogonal basis for L2(R, w(x) dx) is a complete orthogonal system. For an orthogonal
system, completeness is equivalent to the fact that the 0 function is the only function ƒ ∈ L2(R,
w(x) dx) orthogonal to all functions in the system. Since the linear span of Hermite polynomials
is the space of all polynomials, one has to show (in physicist case) that if ƒ satisfies
36. HERMITE POLYNOMIALS
−∞
∞
𝑓(𝑥)𝑥 𝑛
𝑒−𝑥2
dx =0
for every n ≥ 0, then ƒ = 0. One possible way to do it is to see that the entire function
F(z) = −∞
∞
𝑓 𝑥 𝑒 𝑧𝑥−𝑥2
𝑑𝑥 = 𝑛=0
∞ 𝑧 𝑛
𝑛!
𝑓(𝑥)𝑥 𝑛 𝑒−𝑥2
dx =0
vanishes identically. The fact that F(it) = 0 for every t real means that the Fourier transform of ƒ(x)
exp(−x2) is 0, hence ƒ is 0 almost everywhere. Variants of the above completeness proof apply to
other weights with exponential decay. In the Hermite case, it is also possible to prove an explicit
identity that implies completeness (see "Completeness relation" below).
An equivalent formulation of the fact that Hermite polynomials are an orthogonal basis for L2(R,
w(x) dx) consists in introducing Hermite functions (see below), and in saying that the Hermite
functions are an orthonormal basis for L2(R).
Hermite's differential equation
The probabilists' Hermite polynomials are solutions of the differential equation
(𝒆−𝒙 𝟐/𝟐
ù)`+λ 𝒆−𝒙 𝟐/𝟐
u=0
where λ is a constant, with the boundary conditions that u should be polynomially bounded at
infinity. With these boundary conditions, the equation has solutions only if λ is a positive
integer, and up to an overall scaling, the solution is uniquely given by u(x) = Hλ(x). Rewriting
the differential equation as an eigenvalue problem
37. HERMITE POLYNOMIALS
L(u)=Ű−Ú𝑥 = −λ𝑈
solutions are the eigenfunctions of the differential operator L. This eigenvalue problem is
called the Hermite equation, although the term is also used for the closely related equation
Ű−2Ú𝑥 = −2λ𝑈
whose solutions are the physicists' Hermite polynomials.
With more general boundary conditions, the Hermite polynomials can be generalized to
obtain more general analytic functions Hλ(z) for λ a complex index. An explicit formula can
be given in terms of a contour integral (Courant & Hilbert 1953).
Recursion relation
The sequence of Hermite polynomials also satisfies the recursion
𝐻 𝑛+1(x)=x𝐻 𝑛(x)−𝐻′ 𝑛 (x). (probabilist)
𝐻 𝑛+1(x)=2x𝐻 𝑛(x)−𝐻′ 𝑛 (x). (physicist)
38. HERMITE POLYNOMIALS
The Hermite polynomials constitute an Appell sequence, i.e., they are a
polynomial sequence satisfying the identity
𝐻′ 𝑛 (x)=n𝐻 𝑛−1(x) (probabilist)
𝐻′ 𝑛 (x)=2n𝐻 𝑛−1(x) (physicist)
or equivalently
𝐻 𝑛(x+y)= 𝑘=0
𝑛 𝑛
𝑘
𝑥 𝑘
𝐻 𝑛−𝑘(y) (probabilist)
𝐻 𝑛(x+y)= 𝑘=0
𝑛 𝑛
𝑘
𝑥 𝑘 𝐻 𝑛−𝑘(2y)(𝑛−𝑘) (physicist)
Generating function
The Hermite polynomials are given by the exponential generating
function
(probabilist)
39. HERMITE POLYNOMIALS
(physicist)
This equality is valid for all x, t complex, and can be obtained by writing the Taylor expansion
at x of the entire function z → exp(−z2) (in physicist's case). One can also derive the
(physicist's) generating function by using Cauchy's Integral Formula to write the Hermite
polynomials as
Using this in the sum , one can evaluate the remaining integral
using the calculus of residues and arrive at the desired generating function
Expected value
(probabilist)