Notes on Mathematical Methods for Physicists Chapter1

Notes on Mathematical Methods for Physicists $\S1$

Chapter1 Mathematical Preliminaries

$\S1.1$ Infinite Series

$\square$ Comparison Test

Consider a convergent series {$a_n$} , we can use {$a_n$} to study the convergence of series {$u_n$} :

$0\leq u_n \leq a_n \Rightarrow$ {$u_n$} is convergent.

Similarly , consider a divergent series {$a_n$} , we can use {$a_n$} to study the convergence of series {$u_n$} :

$u_n \geq a_n \geq 0 \Rightarrow$ {$u_n$} is divergent.

$\square$ Cauchy Root Test

$(a_n)^{1/n}\leq r < 1 \Rightarrow$ {$a_n$} is convergent.

$(a_n)^{1/n} \geq 1 \Rightarrow$ {$a_n$} is divergent.

$\square$ d'Alembert (or Cauchy) Ratio Test

$\frac{a_{n+1}}{a_n}~\leq r<1 \Rightarrow$ {$a_n$} is convergent.

$\frac{a_{n+1}}{a_n}~\geq 1\Rightarrow$ {$a_n$} is divergent.

$\begin{equation}\underset{n\to\infty}{lim}~(\frac{a_{n+1}}{a_n})\left\{\begin{array}{lr}<1,\quad congvergent &\\=1,\quad divergent &\\>1,\quad indeterminate &\end{array}\right.\end{equation}$

At some crucial point , the test may fail. For example , $a_n=\frac{1}{n}$ (harmonic series) :

but we cannot find r (< 1) independent of n.

Since $\underset{n\to\infty}{lim}\frac{a_{n+1}}{a_n}~=1$ , the test fails.

$\square$ Cauchy (or Maclaurin) Integral Test

Consider $f(x)$ a continuous , monotonic decreasing function , in which $f(n)=a_n$.

We have an equation , it writes ,

proof :

Then $RHS=LHS$.

$Q.E.D\quad\blacksquare$

Alternative of the equation :

In this kind of equation , the second part in the right hand side is a function that oscillates about zero.

$\square$ More Sensitive Tests

1. Kummer Theorem

If $\underset{n\to\infty}{lim}(a_n\frac{u_N}{u_{n+1}}-a_{n+1})\geq C>0$ , where $C$ is a constant , we have {$u_n$} is convergent if $\sum_{n}a_n^{-1}$ is convergent. And if $\sum_{n}a_n^{-1}$ is divergent , the more weak it diverges , the more powerful the theorem will be.

If $\underset{n\to\infty}{lim}(a_n\frac{u_N}{u_{n+1}}-a_{n+1})\leq 0$ , we have {$u_n$} is divergent if $\sum_{n}a_n^{-1}$ is divergent.

proof :

2. Gauss's Test

For large n , $\frac{u_n}{u_{n+1}}=1+\frac{h}{n}+\frac{B(n)}{n^2}$ , we can know that

$\square$ Alternating Series

For series of the form $\sum_{n=1}^{\infty}(-1)^{n+1}a_n$ , $a_n>0$ , we have Leibniz Criterion :

If $a_n$ monotonically decreases , and $lim_{n\to\infty}a_n=0$ , then {$a_n$} converges.

proof :

so when $n\uparrow,R_{2n}\uparrow$.

$Q.E.D\quad \blacksquare$

$\square$ Absolute & Conditional Convergence

Absolute convergence : the absolute value of its terms form a convergent series.

Conditional convergence : not the situation above.

$\square$ Operation on Series

$\bullet$ If an infinite series is absolutely convergent , the series sum is independent of the order in which the terms are added.

$\bullet$ An absolutely convergent series may be added termwise to , or subtracted termwise from , or multiplied termwise with another absolutely convergent series , and the resulting series will also be absolutely convergent.

$\bullet$ The series (as a whole) may be multiplied with another absolutely convergent series. The limit of the product will be the product of the individual series limits. The product series , a double series , will also convergent absolutely.

$\square$ Improvement of Convergence

The rate of convergence : to form a linear combination of our slowly converging series and one or more series whose sum is known.

For the known series , the following collection is particularly useful :

The series we want to sum and one or more known series (multiplied by coefficient) are combined term by term. The coefficients in the linear combination are chosen to cancel the most slowly converging terms.

$\square$ Rearrangement of Double Series

For $S=\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}a_{n,m}$ ,

$\S1.2$ Series of Functions

Let's extend our concept of infinite series to include series of functions :

$\square$ Uniform Convergence

If for any small $\epsilon>1$ , there exists a number $N$ , independent of $x$ in the interval [$a,b$] (that is , $a<x<b$) such that ,

Then the series is said to be uniformly convergent.

$\square$ Weierstrass M (Majorant) Test

If we can construct a series of numbers $\sum_{i=1}^{\infty}M_i$ , in which $M_i\geq|u_i(x)|$ for all $x$ in the interval [$a,b$] , and $\sum_{i=1}^{\infty}M_i$ is convergent , then our series $u_i(x)$ will be uniformly convergent in [$a,b$].

proof :

Uniform convergence has nothing with absolute convergence. But M test can only establish for absolutely convergent series.

$\square$ Abel's Test

A somewhat more delicate test for uniform convergence has been given by Abel. If $u_n(x)$ can be written in the uniform $a_nf_n(x)$ , and

1. The $a_n$ form a convergent series , $\sum_{n}a_n=A$.
2. For all $x$ in [$a,b$] the functions $f_n(x)$ are monotonically decreasing in $n$ , that is , $f_{n+1}(x)\leq f_{n}(x)$.
3. For all $x$ in [$a,b$] , all the $f_n(x)$ are bounded in the range $0\leq f_n(x)\leq M$ , where $M$ is independent of $x$.

Then $\sum_{n}u_n(x)$ converges uniformly in [$a,b$]. This method is especially useful in analyzing the convergence of power series.

$\square$ Properties of Uniformly Convergent Series

If a series $\sum_nu_n(x)$ is uniformly convergent in [$a,b$] and the individual terms $u_n(x)$ are continuous ,

1. The series sum $S(x)=\sum_{n=1}^{\infty}u_n(x)$ is also continuous ,
2. $\int_{a}^{b}S(x)dx=\sum_{n=1}^{\infty}\int_{a}^{b} u_n(x)$ ,
3. if the following additional conditions are also satisfied , then $\sum_{n=1}^{\infty}\frac{du_n(x)}{dx}$ is uniformly convergent in [$a,b$] : $\frac{du_n(x)}{dx}$ is continuous in [$a,b$].

The first and second conditions are always right in physics , but the third is not because it is more restrictive.

$\square$ Taylor's Expansion

We assume that our function $f(x)$ has a continuous n-th derivative in the interval [$a,b$].

First , let's intergrate this n-th derivative n times :

Finally ， after integrating for the n-th time ,

where the remainder , $R_n$ , is given by the n-fold integral ,

We may convert $R_n$ into a perhaps more pratical form by using the mean value theorem of integral calculus ,

Or applying Cauchy's mean value theorem of integral calculus ,

When you adjust n properly , $R_n\to0$. Then we have Taylor Expansion , which writes

$\square$ Power Series

When $a=0$ , we have Maclaurin series ,

$\square$ Properties of Power Series

$f(x)=\sum_{n=0}^{\infty}a_nx^n$ , in which $a_n$ is independent of $x$. If $lim_{n\to\infty}|\frac{a_{n+1}}{a_n}|=R^{-1}$ , then $R$ is the radius of convergence , and the series converges for $x\in(-R,R)$.

But the ratio / root test fails at endpoints , $x=\pm R$ needs special attention. In M test , the series is uniformly and absolutely convergent in ($-R,R$).

$\square$ Uniqueness Theorem

The power-series representation is unique. Assume that

What we need to prove is that $a_n=b_n$ , for all $n$.

When $x=0$ , we have $a_0=b_0$. Then differentiate the series ,

When $x=0$ , we have $a_1=b_1$. ··· ···

Repeating the process , we will get $a_n=b_n$.

$Q.E.D\quad \blacksquare$

This theorem will be a crucial point in our study of differential equations , in which we develop power series solutions (For instance , in theoretical physics , there's perturbation theory in quantum mechanics).

$\square$ Indeterminate Forms

You can use the power-series representation of functions to prove $L'H \hat{o} pital's\,\,\,Rule$

(See exercise 1.2.12)

$\square$ Inversion of Power Series

Consider that $y-y_0=\sum_{n=1}^{\infty}a_n(x-x_0)^n$ , if we want to know $x-x_0=\sum_{n=1}^{\infty}b_n(y-y_0)^n$ , we need to equate coefficients of $(x-x_0)^n$ on both sides of the given equation.

$\S1.3$ Binomial Theorem

Binomial series is a extremely significant application of the Maclaurin series. Let $f(x)=(1+x)^m$ , in which $m\in\R$. Then ,

For this function , the remainder is

with $\xi\in(0,x)$.

When $x>0$ , for $n>m$ , $(1+\xi)^{m-n}$ is a maximum for $\xi>0$. So for $x>0$ , $|R_n|\leq\frac{x^n}{n!}|m(m-1)···(m-n+1)|$ , with $\underset{n\to\infty}{lim}R_n=0$ , when $0\leq x<1$.

Because the radius of convergence of a power series is the same for positive and negative $x$ , the binomial series converges for $-1<x<1$. The endpoints cannot be determined.

Binomial Expansion :

If $m$ is a nonnegative integer , $R_n$ for $n>m$ vanishes for all $x$ , corresponding to the fact that under those conditions $(1+x)^m$ is a finite sum.

Binomial Coefficients :

In which , when $n=0$ , $\begin{equation}\left\lgroup\begin{array}{lr}m\\0\end{array}\right\rgroup =1\end{equation}$.

1. When $m$ is a positive integer , $\begin{equation}\left\lgroup\begin{array}{lr}m\\n\end{array}\right\rgroup =\frac{m!}{n!(m-n)!}\end{equation}$. That is $C_m^n=\begin{equation}\left\lgroup\begin{array}{lr}m\\n\end{array}\right\rgroup\end{equation}$.

2. For negative integer $m$ , set $m=-p$ , there is

3. For nonintegral $m$ , it is convenient to use the Pochhammer Symbol , defined for general a and nonnegative integer $n$ , as

For addition , $0!!=(-1)!!=1$ , because $(2n)!!=2^nn!$ , and $(2n-1)!!=\frac{(2n)!!}{2^nn!}$.

$\square$ Generalized Binomial Expansion

1. For positive integer $n$ , to polynomials ,

In which $\sum_{i=1}^mn_i=n$.

$\S1.4$ Mathematical Induction

If a relation is valid for an arbitrary value of some index $n$ , then it is also valid if $n$ is replaced by $(n+1)$.

$\S1.5$ Operations on Series Expansions of Functions

For example ,

$\S1.6$ Some Important Series

$\S1.7$ Vectors

Scalar is defined as quantity that has algebraic magnitude only.

Vector is defined as quantity that has magnitude and direction. Vectors defined over a region are called vector fields.

$\square$ Basic Properties

Let's just skip this part ···

$\square$ Dot (Scalar) Product

Algebraic formula : $\vec{A}\cdot\vec{B}=\sum_iA_iB_i=\sum_iB_iA_i=\vec{B}\cdot\vec{A}$

( That is , if $\vec{C}=\vec{A}+\vec{B}$ , then $\vec{A}\cdot\vec{B}=\frac{1}{2}(|\vec{C}|^2-|\vec{A}|^2-|\vec{B}|^2)$ )

Geometric formula : $\vec{A}\cdot\vec{B}=|\vec{A}||\vec{B}|cos\theta$ , in which $\theta=<\vec{A},\vec{B}>$.

The projection of $\vec{A}$ in direction of $\vec{B}$ : $\vec{A_b}=(\vec{A}\cdot\frac{\vec{B}}{|\vec{B}|})\frac{\vec{B}}{|\vec{B}|}$

Schwarz inequality : $|\vec{A}\cdot\vec{B}|\leq|\vec{A}||\vec{B}|$

$\square$ Orthogonality

If $\vec{A}、\vec{B}\neq0$ , then

$\S1.8$ Complex Numbers & Functions

$\square$ Basic Properties

A complex number is an ordered pair of two real variables , $z=(x,y)$ (the order is significant) , in which $i=(0,1)$ is the imaginary unit.

Addition : $z_1+z_2=(x_1,y_1)+(x_2,y_2)=(x_1+x_2,y_1+y_2)$

Multiplication : $z_1\cdot z_2=(x_1,y_1)\cdot(x_2,y_2)=(x_1x_2-y_1y_2,x_1y_2+x_2y_1)$

For historical reasons , $i$ and its multiples are known as imaginary numbers.

The space of complex numbers , sometimes denoted $\Z$ , by mathematicians , has the following formal properties :

1. It is closed under addition and multiplication , meaning that if two complex numbers are added or multiplied , the result is also a complex number.

2.It has a unique zero number , which when added to any complex number leaves it unchanged and which , when multiplied with any complex number yields zero.

3.It has a unique unit number $1$ , which when multiplied with any complex number leaves it unchanged.

4.Every complex number $z$ has an inverse under addition (known as $-z$) , and every nonzero $z$ has an inverse under multiplication , denoted $z^{-1}$ or $1/z$.

5.It is closed under exponentiation : if $u$ and $v$ are complex numbers , the $u^v$ is also a complex number.

Complex conjugation : $z=x+iy\Longleftrightarrow z^*=x-iy$

Division :

$\square$ Functions in the Complex Domain

Among other things , that if a function is represented by a power series , we should , within the region of convergence of the power series , be able to use such series with complex values of the expansion variable. This is called the permanence of the algebraic form.

which is called Euler equation.

when $z=x+iy$ , then $f(z)=u(x,y)+iv(x,y)$ has real part $\mathscr{Re}(f(z))=u(x,y)$ and imaginary part $\mathscr{Im}(f(z))=v(x,y)$.

$\square$ Polar Representation

Skip that $\cdots$

$z=re^{i\theta}$ , $r$ is called modulus , $\theta$ is called argument or phase. Since additon on an Argand diagram is analogous to $2-D$ vector addition , it can be seen that

Remember , $w(z)$ is a mapping from $xy$ plane to $uv$ plane.

$\square$ Complex Numbers of Unit Magnitude

They are on the unit circle.

$\square$ Circular & Hyperbolic Functions

$\square$ Power & Root

The polar form is very convenient for expressing powers and roots of complex numbers. Consider when $z=re^{i\varphi}$ , then

Actually , $1/n$ root has $n$ different values.

$\square$ Logarithm

$\Longrightarrow$ $ln(z)$ has infinite number of values.

$\S1.9$ Derivatives & Extrema

Definition of derivative :

Variation $\&$ differential : $df=\frac{df}{dx}dx$

When $f(x,y,z)$ , we have $df=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz$ , in which $\frac{\partial f}{\partial x}$ is called partial derivatives.

Cross derivatives : $\frac{\partial^2 f}{\partial x\partial y}=\frac{\partial^2 f}{\partial y\partial x}$

1.The chain rule :

2.Consider $f=f(x,y)$ , then :

In Lagrangian mechanics , we often encounter formulas such as

In which we need to distinguish between $\frac{d}{dt}$ and $\frac{\partial}{\partial t}$.

$\square$ Stationary Points

If we want to know how a function $f$ changes if we move in various directions in the space of the independent variables , we can use

In which the direction is given by $\frac{dx_i}{ds}$.

It is often to find the minimum of a function $f$ of $n$ variables $x_i$ , $i=1,2,\cdots,n$ , and a necessary but not sufficient condition on its position is that $\frac{df}{ds}=0$ , for $ds$ in any direction. And this equals to $\frac{\partial f}{\partial x_i}=0$ , $i=1,2,\cdots,n$.

All points that satisfies the formula above are termed stationary :

$\S1.10$ Evaluation of Integrals

$\star\star\star$ Proficiency in the evaluation of integrals involves a mixture of experience , skill in pattern recognition , and a few tricks. $\star\star\star$

$\square$ Integration by Parts

Legendre Transformation :

$\square$ Special Functions

Functions	Definitions	Addition
Gamma Function	$\Gamma(x)=\int_0^{\infty}t^{x-1}e^{-t}dt$	See Chapter 13.
Factorial ($n$ integral)	$n!=\Gamma(n+1)$	/
Riemann Zeta Function	$\zeta(x)=\frac{1}{\Gamma(x)}\int_0^\infty\frac{t^{x-1}dt}{e^t-1}$	See Chapter 1.$\&$12.
Exponential Integrals	$E_n(x)=\int_1^\infty t^{-n}e^{-xt}dt$	$=\int_x^\infty t^{-n}e^{-t}dt$
	$Ei(x)=\int_{-\infty}^{x}t^{-1}e^{-t}dt$	$E_1(x)=-Ei(-x)$
Sine Integral	$Si(x)=-\int_x^{\infty}\frac{sin(t)}{t}dt$	/
Cosine Integral	$Ci(x)=-\int_x^{\infty}\frac{cos(t)}{t}dt$	/
Error Functions	$erf(x)=\frac{2}{\sqrt{\pi}}\int_0^xe^{-x^2}dt$	$erf(\infty)=1$
	$erfc(x)=\frac{2}{\sqrt{\pi}}\int_x^\infty e^{-x^2}dt$	$erfc(x)=1-erf(x)$
Dilogarithm	$Li_2(x)=-\int_0^x\frac{ln(1-t)}{t}dt$	/

$\square$ Other Methods

An extremely powerful method for the evaluation of difinite integrals is that of contour integration in the complex plane. But this method will be presented in Chapter11. and not be dicussed here.

1.Integrals can often be evaluated by methods that involve integration or differentiation with respect to parameters , thereby obtaining relations between known integrals and those whose values are being sought.

2.Many integrals can be evaluated by first converting them into infinite series , then manipulating the resulting series , and finally either evaluating the series or recognizing it as a special function.

3.Simply using complex numbers aids in the evaluation of some integrals.

4.Recursion is useful in obtaining formulas for a set of related integrals.

$\square$ Multiple Integrals

In addition to the techniques available for integration in a single variable , multiple integrals provide further opportunities for evaluation based on changes in the order of integration and in the coordinate system used in the integral.

A significant change in the form of the $2-D$ or $3-D$ integrals can sometimes be accomplished by changing between Cartesian and polar coordinate systems.

$\square$ Remarks : Changes of Integration Variables

In a $1-D$ integration , a change in the integration variable from , say , $x$ to $y(x)$ involves two adjustments :

1.$dx\longrightarrow(\frac{dx}{dy})dy$

2.$x_1,x_2\longrightarrow y(x_1),y(x_2)$

If $y(x)$ and $x(y)$ is not single-valued , the process becomes more complicated , and we will not discuss it here.

For multiple integrals , we use Jacobian. For $dxdy\longrightarrow Jdudv$ , $J=\frac{\partial(x,y)}{\partial(u,v)}$.

The computation of Jacobian will be discussed in Section4.4. Here , remember to determinate the transformed region.

$\S1.11$ Dirac Delta Function

Definition :

No such function exists , (in the usual sense). However , the crucial property can be developed rigorously as the limit of a sequence of functions. The common seen examples are as follows :

Examples	Properties
$\begin{equation}\delta_n(x)=\left\{\begin{array}{lr}0\,,\,(-\infty,-\frac{1}{2n})\cup(\frac{1}{2n},\infty)\\n\,,\,(-\frac{1}{2n},\frac{1}{2n})\end{array}\right.\end{equation}$	easy to integrate
$\delta_n(x)=\frac{n}{\sqrt{\pi}}e^{-n^2x^2}$	its derivatives leads to Hermite Polynomials
$\delta_n(x)=\frac{n}{\pi}\frac{1}{1+n^2x^2}$	/
$\delta_n(x)=\frac{sin(nx)}{\pi x}=\frac{1}{2\pi}\int_{-n}^ne^{-ixt}dt$	Fourier analysis or quantum mechanics
	Dirichlet Kernel : $\frac{sin[(n+\frac{1}{2})x]}{2\pi sin(\frac{1}{2}x)}$

They are of different uses as above.

For most physical purposes , the forms describing delta functions are quite adequate. However, from a mathematical point of view , the limits $\underset{n\to\infty}{lim}\delta_n(x)$ do not exist. To avoid the difficulty , label $\delta(x)$ a distribution , and write :

$\square$ Properties of $\delta(x)$

$\bullet$ $\delta(-x)=\delta(x)$

$\bullet$ $\delta(ax)=\frac{1}{|a|}\delta(x)$

Proof :

$\bullet$ $\int_{-\infty}^{\infty}f(x)\delta(x-x_0)dx=f(x_0)$

$\bullet$ If the argument of $\delta(x)$ is a function $g(x)$ with simple zeros at points $a_i$ on the real axis (and therefore $g'(a_i)\neq0$) , then

Proof :

$\bullet$ Derivative of delta function :

This is the definition of $\delta'(x-x_0)$.

$\bullet$ In three dimensions , the delta function $\delta(\vec{r})$ is intepreted as $\delta(x)\delta(y)\delta(z)$ , irrespective of the coordinate system in use. Thus , in spherical polar coordinates ,

$\bullet$ $\delta(t-x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{i\omega(t-x)}d\omega$ (See Chapter20. , Fourier Integrals)

$\bullet$ Expansions of $\delta(x)$ are addressed in Chapter5. (Example5.1.7).

$\square$ Kronecker Delta

The discrete analog of the Dirac delta function ,

Usage examples :