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MM311: Mathematics 4B
by Martin Grant
# MM311: Mathematics 4B

## 0. Probability and Statistics

### 53.202 Materials

### Presentation and Summarisation of Data

### RandomVariables

### Basic Laws

### Probability Theory

### Measures of Spread

### Measures of Locaion

### Z-test, samples and confidence

## Coursework (Semester 2)

### Sheet 1

### Sheet 2

### Sheet 3

## 1. Fourier Series

### Periodic Functions

### Extended Functions

### Sketching Functions

### Odd and Even Functions

### Half Range Fourier Series

### Continuous Fourier Series

### Term-by-Term Differentiation

### Complex Exponential Form

### Tutorials

### Worked Examples

## 2. Partial Differential Equations

### Solving Simple PDE's

### Change of Variables

### Real Examples of PDE's

### Linearity, Homogeneity and Superposition

### Constant Coefficient ODE's

### Separation of Variables

### Tutorials

### Worked Examples

## 3. Complex Analysis

### Complex Numbers

### Points in the Complex Plane

### Complex Functions

### Continuity

### Analytic Functions

### The Cauchy-Riemann Equations

### Harmonic Functions

### Functions of Complex Variables

### Line Integrals in the Complex Plane

### Parameterising Curves

### Cauchy's Integral Theorem

### Cauchy's Integral Formula

### Tutorials

## 4. Techniques of Complex Integration

### Taylor and Laurent Series

### Residue Integration

### Real Integration using Residue Theorem

### Tutorials

## 5. Fourier Transforms

### The Fourier Transform

### The Fourier Sine and Cosine Transforms

### Properties of the Fourier Transform

### Parseval's Identity

### Dirac Delta Function

### Generalised Fourier Transforms

### Application of Residue
Integration to Fourier Transforms

### Tutorials

## Sample Papers

### Sample Paper 1

### Sample Paper 2

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Lecture Notes

Worked Examples

Tutorials and Solutions

Past Papers

Hisograms

Stem and Leaf Plots

Box Plots

Good and Bad Presentation

Introduction to Exploratory Data

Lecture Slides, Lecture 1, Lecture 2

General Results

Discrete and Continuous Random Variables

Expectation

Variance

Moments

Quartiles

Mean and Variance of Linear Combinaions

Lecture Slides, Lecture 5, Lecture 6

P(A or B)

P(not A)

P(A and B)

P(A|B)

Independence

Bayes Theorem

Lecture Slides, Lecture 4, Lecture 5

Introduction

Origin of Probability

Assignment of Probability

Disributions, Poisson, Binomial, Geometric, Negative ~Exponential, Nomal

Lecture Slides, Lecture 3, Lecture 7, Lecture 8, Lecture 9

Variance

Standard Deviation

Range

Quartiles

Semi-interquartile Range

Lecture Slides, Lecture 6

Mean

Median

Mode

Lecture Slides, Lecture 10, Lecture 11, Lecture 12

Solutions

Solutions

Solutions

Fourier Sine Series

Fourier Cosine Series

Solutions

Heat Equation

Wave Equation

Solutions

Complex Exponential

Complex Trig & Hyperbolic Functions

Complex Logarithm

CIF for Derivatives of Analytic Functions

Solutions (Part 1)

Solutions (Part 2)

Poles

Method

Theorem

Residue at Simple Pole, 1st Method, 2nd Method

Residue at nth Order Pole

Integrals of Rational Functions

Improper Integrals

Fourier Integrals

Solutions

Linearity

Derivative

Time-Shift

Frequency-Shift

Duality

Convolution of Two Functions, In Time, In Frequency

Multiplication

Solutions

Solutions