Maths for Physics and Astronomy Complete Course Notes

Maths for Physics and Astronomy Course Overview

• Course Code: MATH0043

• Level: 5 (Undergraduate)

• Credit Value: 15 credits (7.5 ECTS credits)

• Term: 2

• Assessment Breakdown: 85% examination, 15% coursework

• Normal Pre-requisites: PHAS0025

• Lecturers: A. Pokrovskiy and J. Hofmann

• Course Description:

  • This course focuses on advanced mathematical methods for Physics and Astronomy students intending to engage in more theoretical studies.

  • It assumes a foundational understanding of basic mathematics and serves as a precursor to a third-year mathematics course.

  • Emphasis is placed on practical understanding rather than strict mathematical rigor.

• Recommended Texts:

  • Boas, Mathematical Methods in the Physical Sciences (Wiley).

  • Wylie and Barrett, Advanced Engineering Mathematics (McGraw-Hill).

Detailed Syllabus

Complex Variables

• Functions of a Complex Variable:

  • Revision of complex numbers and power series.

  • Elementary functions: logarithmic function, fractional powers, branch points, and cuts.

  • Concepts of continuity and differentiability: analytic functions, Cauchy-Riemann equations, harmonic functions.

  • Types of singularities: Cauchy’s Theorem, Residue Theorem.

  • Evaluation of integrals: Cauchy’s integral formulae, evaluation of real integrals.

Calculus of Variations

• Euler’s Equation

  • Extremal of a functional involving one dependent variable.

  • Multiple examples with one dependent variable.

  • Problem-solving with integral constraints.

Group Theory

• The Group Axioms:

  • Examples of symmetries for finite groups.

  • Definition and identification of subgroups.

  • Examples of representations of finite groups: similarity transformations.

  • Reducible and irreducible representations.

  • Understanding of characters.

Week 1: Complex Numbers - Part I

Introduction to Complex Numbers

• Definition: Complex numbers are represented as $z = x + iy$ where:

  • $x, y ext{ are real numbers, } ext{ } (x, y ext{ } extless ext{ } extbf{C})$.

• Imaginary Unit: $i = ext{the square root of } -1$, satisfying the property $i^2 = -1$.

• Complex Conjugate: $z̄ = x - iy$.

• Operations among complex numbers include addition, subtraction, multiplication, and division derived from the definition of $i^2$.

Modulus and Argument of a Complex Number

• Modulus: The modulus of a complex number $|z| = r = ext{ } ext{the distance from the origin}$ in the Argand diagram:
  $|z| = ext{ } ext{Sqrt}(x^2 + y^2).$

• Argument: The argument $Arg(z) = θ = an^{-1} rac{y}{x}$; by convention defined as
  -π < Arg(z) ≤ π.

• Trigonometric relationships:

  • $x = r ext{cos} θ$

  • $y = r ext{sin} θ $

  • Hence, $z = x + iy = r( ext{cos} θ + i ext{sin} θ)$.

• Polar Form: Introduced as $z = re^{iθ}$.

Taylor Series Expansion of Exponential Functions

• Euler’s Identity: $e^{iθ} = ext{cos} θ + i ext{sin} θ.$

• DeMoivre’s Theorem: This leads to expressing powers of complex numbers:

  • $z^n = (e^{iθ})^n = e^{inθ} ext{ (for } r = 1)$.

• Example: Find $ ext{cos} 5θ$ in terms of powers of $ ext{cos} θ$:

  • $ext{cos} 5θ = 16 ext{cos}^5 θ - 20 ext{cos}^3 θ + 5 ext{cos} θ.$

Complex Functions and Analyticity

• Complex Function: Defined as $w = f(z)$ mapping from $ extbf{C}
  ightarrow extbf{C}$, expressed as
  $w = f(z) = u(x, y) + iv(x, y),$

• Functions are termed continuous if nearby points in the z-plane map to nearby points in the w-plane.

  • Example: Mapping the function $w = f(z) = z^2$ and analyzing the mapping of selected points on the Argand diagram.

  • Key property illustrated: $|w| = |z|^2 ext{ and } Arg(w) = 2Arg(z).$

Multi-valued Functions

• Example: For $w = f(z) = z^{1/2}$, the complex plane can exhibit multi-valued characteristics where each point maps to two points in the w-plane, necessitating careful consideration of branch points and cuts.

• Branch Point: A point $z_0$ is a branch point if there exists a discontinuity of the function as a circuit around that point is traversed.

Branch Cuts

• Defined to ensure that a multi-valued function can be treated as a single-valued function. Example provided detailing the branch cut along the negative real axis.

Week 2: Complex Calculus

Complex Power Series

• Definition of convergence and absolute convergence.

• Presentation of complex power series defined as $ext{Series: } ext{ } ext{ } ∑<em>{n=0}^{∞} a</em>n(z - z_0)^n.$

  • Radius of convergence is defined, linking to the ratio test.

Differentiation of Complex Functions

• Definition of differentiability in the context of complex analysis stressing independence of direction of approach.

  • Holomorphic functions will follow Cauchy-Riemann equations leading to detailed handling of differentiability.

Key Theorems and Integrals

Cauchy-Riemann Equations

• Integral in the field:

  • $rac{ ext{∂u}}{ ext{∂x}} = rac{ ext{∂v}}{ ext{∂y}}, ext{ } rac{ ext{∂u}}{ ext{∂y}} = - rac{ ext{∂v}}{ ext{∂x}}$.

Cauchy’s Theorem

• Integral stating if $f(z)$ is analytic in D, then
  $ext{Circumference integral } ∑_C f(z) dz = 0$.

Three Vital Aspects for Integration

1. Single-Valued Functions: The importance in evaluating functions lies in their uniqueness.

2. Path Independence: Integration results are reliant on the analytic nature of the functions leading to simplified closed contours not yielding contribution.

3. Residue Theorems: Residues of complex integrals lead to valuable computations, often yielding real-valued integral results.

Practical Applications

Evaluation of Integrals Using Residue Theorem

• Residue sinking parameters leading to concise analytic method for evaluating integrals such as $I = ∫^{∞}_{−∞} f(x) dx$.

  • Formulation navigates through both theoretical necessity applied in practical scenarios, mapping and calculating integrated outputs in real-time applications.

Final Results and Applications

• Examples provided throughout the syllabus will demonstrate the application through practical scenarios and analytical derivations.

Conclusion

• Comprehensive analysis over topics of complex integration, group theory, calculus of variations, and other integral calculus elements, coupled with clear definitions and proofs aims to furnish students with a robust grasp of the material for application in Physics and Astronomy fields.