1. 0. Probability and Statistics
1.1. 53.202 Materials
1.1.1. Lecture Notes
1.1.2. Worked Examples
1.1.3. Tutorials and Solutions
1.1.4. Past Papers
1.2. Presentation and Summarisation of Data
1.2.1. Hisograms
1.2.2. Stem and Leaf Plots
1.2.3. Box Plots
1.2.4. Good and Bad Presentation
1.2.5. Introduction to Exploratory Data
1.2.6. Lecture Slides
1.2.6.1. Lecture 1
1.2.6.2. Lecture 2
1.3. RandomVariables
1.3.1. General Results
1.3.2. Discrete and Continuous Random Variables
1.3.3. Expectation
1.3.4. Variance
1.3.5. Moments
1.3.6. Quartiles
1.3.7. Mean and Variance of Linear Combinaions
1.3.8. Lecture Slides
1.3.8.1. Lecture 5
1.3.8.2. Lecture 6
1.4. Basic Laws
1.4.1. P(A or B)
1.4.2. P(not A)
1.4.3. P(A and B)
1.4.4. P(A|B)
1.4.5. Independence
1.4.6. Bayes Theorem
1.4.7. Lecture Slides
1.4.7.1. Lecture 4
1.4.7.2. Lecture 5
1.5. Probability Theory
1.5.1. Introduction
1.5.2. Origin of Probability
1.5.3. Assignment of Probability
1.5.4. Disributions
1.5.4.1. Poisson
1.5.4.2. Binomial
1.5.4.3. Geometric
1.5.4.4. Negative ~Exponential
1.5.4.5. Nomal
1.5.5. Lecture Slides
1.5.5.1. Lecture 3
1.5.5.2. Lecture 7
1.5.5.3. Lecture 8
1.5.5.4. Lecture 9
1.6. Measures of Spread
1.6.1. Variance
1.6.2. Standard Deviation
1.6.3. Range
1.6.4. Quartiles
1.6.5. Semi-interquartile Range
1.6.6. Lecture Slides
1.6.6.1. Lecture 6
1.7. Measures of Locaion
1.7.1. Mean
1.7.2. Median
1.7.3. Mode
1.8. Z-test, samples and confidence
1.8.1. Lecture Slides
1.8.1.1. Lecture 10
1.8.1.2. Lecture 11
1.8.1.3. Lecture 12
2. 1. Fourier Series
2.1. Periodic Functions
2.2. Extended Functions
2.3. Sketching Functions
2.4. Odd and Even Functions
2.5. Half Range Fourier Series
2.5.1. Fourier Sine Series
2.5.2. Fourier Cosine Series
2.6. Continuous Fourier Series
2.7. Term-by-Term Differentiation
2.8. Complex Exponential Form
2.9. Tutorials
2.9.1. Solutions
2.10. Worked Examples
3. 4. Techniques of Complex Integration
3.1. Taylor and Laurent Series
3.1.1. Poles
3.2. Residue Integration
3.2.1. Method
3.2.2. Theorem
3.2.3. Residue at Simple Pole
3.2.3.1. 1st Method
3.2.3.2. 2nd Method
3.2.4. Residue at nth Order Pole
3.3. Real Integration using Residue Theorem
3.3.1. Integrals of Rational Functions
3.3.2. Improper Integrals
3.3.3. Fourier Integrals
3.4. Tutorials
3.4.1. Solutions
4. 5. Fourier Transforms
4.1. The Fourier Transform
4.2. The Fourier Sine and Cosine Transforms
4.3. Properties of the Fourier Transform
4.3.1. Linearity
4.3.2. Derivative
4.3.3. Time-Shift
4.3.4. Frequency-Shift
4.3.5. Duality
4.3.6. Convolution of Two Functions
4.3.6.1. In Time
4.3.6.2. In Frequency
4.3.7. Multiplication
4.4. Parseval's Identity
4.5. Dirac Delta Function
4.6. Generalised Fourier Transforms
4.7. Application of Residue Integration to Fourier Transforms
4.8. Tutorials
4.8.1. Solutions
5. Coursework (Semester 2)
5.1. Sheet 1
5.1.1. Solutions
5.2. Sheet 2
5.2.1. Solutions
5.3. Sheet 3
5.3.1. Solutions
6. 2. Partial Differential Equations
6.1. Solving Simple PDE's
6.2. Change of Variables
6.3. Real Examples of PDE's
6.3.1. Heat Equation
6.3.2. Wave Equation
6.4. Linearity, Homogeneity and Superposition
6.5. Constant Coefficient ODE's
6.6. Separation of Variables
6.7. Tutorials
6.7.1. Solutions
6.8. Worked Examples
7. 3. Complex Analysis
7.1. Complex Numbers
7.2. Points in the Complex Plane
7.3. Complex Functions
7.4. Continuity
7.5. Analytic Functions
7.6. The Cauchy-Riemann Equations
7.7. Harmonic Functions
7.8. Functions of Complex Variables
7.8.1. Complex Exponential
7.8.2. Complex Trig & Hyperbolic Functions
7.8.3. Complex Logarithm
7.9. Line Integrals in the Complex Plane
7.10. Parameterising Curves
7.11. Cauchy's Integral Theorem
7.12. Cauchy's Integral Formula
7.12.1. CIF for Derivatives of Analytic Functions
7.13. Tutorials
7.13.1. Solutions (Part 1)
7.13.2. Solutions (Part 2)
8. Sample Papers
8.1. Sample Paper 1
8.1.1. Solutions