Problems and Solutions in Theoretical and Mathematical Physics

The following sections are included:

• Preface

• Contents

• Notation

A complex number z can be written as z = x + iy, x, y ∈ ℝ where x is the real part ℜ(z), y is the imaginary part ℑ(z) and

Arithmetic series, geometric series and the harmonic series play a role in problems in theoretical and mathematical physics. An arithmetic series is the sum of a sequence { s_k }, k = 0, 1, … in which each term is calculated from the previous one by adding (or subtracting) a constant c. This means

s_k = s_k−1 + c = s_k−2 + 2c = … = s₀ + c · k

where k = 1, 2, …. A geometric series is a series with a constant ratio between sucessive terms…

Let N be a positive integer, N ≥ 2. The discrete one-dimensional Fourier transform for a given sequence

x(n), n = 0, 1, 2, …,N − 1

is defined by

A complex polynomial is a complex function of the form

p(z)= c_nzⁿ + c_n−1zⁿ⁻¹ + … + c₀

where c₀, c₁,… c_n are complex numbers and n is a natural number. A root, or zero, of this polynomial is a complex number w such that p(w) = 0. The fundamental theorem of algebra states that any complex polynomial must have a complex root…

Let ℱ be a field, for example the set of real numbers ℝ or the set of complex numbers ℂ. Let m, n ≥ 1 be two integers. An array A of numbers in ℱ

A group G is a set of objects {a, b, c,…} (not necessarily countable) together with a binary operation which associates with any ordered pair of elements a, b in G a third element ab in G (closure)…

Let A be an n × n matrix over ℂ. The eigenvalue equation is given by

Ax = λx

with x ≠ 0. Here λ is called the eigenvalue and x is called the corresponding eigenvector. Thus for any eigenvector x of A, Ax is a scalar multiple of x…

Problem 1. Let A and L be two n x n matrices. Calculate

e^LAe^–L…

Problem 1. Let ƒ:[0,1] ↦ [0,1] be defined by

ƒ(x) =4x(1 – x)

and let ∅ :[0,1] ↦ [0,1] be defined by …

Suppose that the functions g and h are continuously differentiable and dh/dx ≠ 0 on an open interval I that contains a (except possibly at a). Suppose that

Let M be a smooth manifold and f be a real valued function of class C⁽¹⁾ on some open set containing M. We consider the problem of finding the extrema of the function ƒ|M. This is called a problem of constrained extrema…

The superposition principle applies to linear difference equations. If two solutions of the linear difference equation are given then the sum of the two solutions is also a solution of the linear difference equation. A solution can also be multiplied with a constant to be a solution again…

The superposition principle applies to linear differential equations. If two solutions of the linear differential equation are given then the sum of the two solutions is also a solution of the linear differential equation. A solution can also be multiplied with a constant to be a solution again…

The fundamental theorem of calculus states that if a function f is Riemann integrable on the interval I = [a; b] and g is a differentiable function on (a, b) such that dg/dx = f(x) for all x ∈ (a, b) and lim_x→a + g(x), lim_x→b − g(x) exist, then

Let L₂(ℝ) be the Hilbert space of the square integrable function over ℝ. Let f ∈ L₂(ℝ) and ƒ ∈ L₁(ℝ). The Fourier transform of f(x) is defined as

Both Cauchy's integral formula and the residue theorem play a central role in solving problems in complex analysis.

Cauchy's Integral Formula. Let Γ be a simple closed positively oriented contour. If f is analytic in some simply connected domain D containing Γ and z₀ is any point inside Γ, then

Legendre, Hermite, Laguerre and Chebyshev polynomials play an important role in mathematical physics, for example as an orthonormal basis in the Hilbert space of square integrable functions. Jacobi elliptic functions and Weierstrass elliptic functions play a role in the solution of nonlinear differential equations…

In a normed vector space V, for example ℂⁿ, the triangular inequality is ‖v + w‖ ≤ ‖v‖ + ‖w‖ for all v,w ∈ V. Besides the triangular inequality the Cauchy-Schwarz, Hölder's, Bessel's, Parseval's, Gibb's and Jensen's inequalities play a central role in mathematical and theoretical physics…

Let F be a closed set on a complete metric space X. A contracting mapping is a mapping ƒ:F→F such that …

In combinatorics we study the enumeration, combination, and permutations of sets of elements …

A subset C of ℝⁿ is said to be convex if for any a and b in C and any θ in ℝ, 0 ≤ θ ≤ 1, the n-tuple θa + (1 − θ)b also belongs to C. In other words, if a and b are in C, then

{θ + (1 − θ)b : 0 ≤ θ ≤ 1} ⊂ C…

Problem 1. A number of linear optimization problems may be cast in the form

The following sections are included:

• Bibliography

• Index