# Notes on Mathematical Methods for Physicists Chapter1

本文最后更新于 2024年5月22日 晚上

# Notes on Mathematical Methods for Physicists

## Chapter1 Mathematical Preliminaries

Infinite Series

Comparison Test

Consider a convergent series {

Similarly , consider a divergent series {

Cauchy Root Test

d'Alembert (or Cauchy) Ratio Test

At some crucial point , the test may fail. For example ,

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

Since

Cauchy (or Maclaurin) Integral Test

Consider

We have an equation , it writes ,

proof :

Then

Alternative of the equation :

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

More Sensitive Tests

##### 1. Kummer Theorem

If**And ifis divergent , the more weak it diverges , the more powerful the theorem will be.**

If

proof :

##### 2. Gauss's Test

For large n ,

Alternating Series

For series of the form

**Ifmonotonically decreases , and, then {} converges.**

proof :

so when

Absolute & Conditional Convergence

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

Conditional convergence : not the situation above.

Operation on Series

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.

Rearrangement of Double Series

For

Series of Functions

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

Uniform Convergence

If for any small

Then the series is said to be **uniformly convergent**.

Weierstrass M (Majorant) Test

If we can construct a series of numbers**uniformly convergent** in [

proof :

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

Abel's Test

A somewhat more delicate test for uniform convergence has been given by Abel. If

- The
form a convergent series , . - For all
in [ ] the functions are monotonically decreasing in , that is , . - For all
in [ ] , all the are bounded in the range , where is independent of .

Then

Properties of Uniformly Convergent Series

If a series

- The series sum
is also continuous , , - if the following additional conditions are also satisfied , then
is uniformly convergent in [ ] : is continuous in [ ].

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

Taylor's Expansion

We assume that our function

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

Finally ， after integrating for the n-th time ,

where the remainder ,

We may convert

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

When you adjust n properly ,**Taylor Expansion** , which writes

Power Series

When

Properties of Power Series

But the ratio / root test fails at endpoints ,

Uniqueness Theorem

**The power-series representation is unique.** Assume that

What we need to prove is that

When

When

Repeating the process , we will get

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).

Indeterminate Forms

You can use the power-series representation of functions to prove

(See exercise 1.2.12)

Inversion of Power Series

Consider that

Binomial Theorem

Binomial series is a extremely significant application of the Maclaurin series. Let

For this function , the remainder is

with

When

Because the radius of convergence of a power series is the same for positive and negative

**Binomial Expansion :**

If

**Binomial Coefficients :**

In which , when

1. When

2. For negative integer

3. For nonintegral**Pochhammer Symbol** , defined for general a and nonnegative integer