2019학년도 1학기

반도체 공학과 공학수학1

주교재: Erwin Kreyszig, Engineering Mathematics, 10th Edition

부교재: 이상구 외 4인, 최신공학수학 I,  1st Edition

강의시간:  공학수학1, (화09:00-10:15)   (목10:30-11:45)

담당교수:  김응기 박사

Week 7

7주차

5.4 Bessel’s Equation. Bessel Functions

*5.5 Bessel Functions of the . General Solution

7.5 베셀 방정식

5.4 Bessel’s Equation. Bessel Functions

Bessel’s equation

The equation is written in standard from (obtained by dividing by )

.

Accordingly, we substitute the series

The derivatives is

We obtain

We find

From we obtain the indicial equation by dropping ,

(4)

The roots are and .

(i)

From

From we give recursion formula

.

so

so

so

so

so

...

and .

The series solution is

.

.

For , The series is a convergence in the interval .

(ii)

From we give recursion formula

.

Replacing by by .

For , is .

By choosing .

If is not an integer, and are linearly independent and the general solution is

( are an arbitrary constants).

The radius of convergence of the series solution is .

Bessel Functions For Integer

We have to make a choice of .

(9)

is called the Bessel function of order

is called the Bessel function of the first kind of order .

Example 1  Bessel Functions and .

Solution

The Bessel function of order is .

The Bessel function of order is .

Sage Coding

 # (주의) bessel 관련 아래 코드는 https://sagecell.sagemath.org/  에서 실행하세요. f = bessel_J(0, x)                 # Bessel 함수 정의 g = bessel_J(1, x) h = bessel_J(2, x) print 'J0(x):', f.taylor(x, 0, 6)     # 급수의 일부항만 제시 print 'J1(x):', g.taylor(x, 0, 8)  print 'J2(x):', h.taylor(x, 0, 6)  plot(f, (x, 0, 10)) + plot(g, (x, 0, 10), color = 'red') + plot(h, (x, 0, 10), color = 'green')

Evaluate

J0(x): -1/2304*x^6 + 1/64*x^4 - 1/4*x^2 + 1

J1(x): -1/18432*x^7 + 1/384*x^5 - 1/16*x^3 + 1/2*x

J2(x): 1/3072*x^6 - 1/96*x^4 + 1/8*x^2

Gamma Function

The gamma function defined by the integral

.

By integration by parts we obtain

This yields the basic function relation

(16)        .

In particular

.

is positive integer

In general

(17)

This shows the gamma function does in fact generalize the factorial function. Now in we had . This is by .

It suggests to choose for any ,

(18)

Then becomes

But gives in the denominator

,

and so on, so that

.

Hence because of our choice of the coefficients simply are

(19)         .

With these coefficients and we get from a particular solution of , denoted by and given by

(20)

is called the Bessel function of the first kind of order .

General Solution for Noninteger . Solution

For not an integer this is easy. Replacing by in

(20)

Theorem  General Solution of Bessel’s Equation

If is not an integer, a general solution of Bessel’s equation for all is

(22)      ( are an arbitrary constant)

If is an integer, then is not a general solution because of linear dependence.

Theorem  Linear Dependence of Bessel Function and

For integer the Bessel function and are linearly dependent, because

.

Proof

(Substituting into )

.

Now since is infinite for , the first sum on the right is or negative integer.

We obtain

.

Theorem  For positive integer the Bessel function and are

Proof

is an even, then is an even function, is an odd, then is an odd function.

Theorem 1  Derivative, Recursion

The derivative of with respect to can be expressed by or by the formulas

(1)                 (2)

Furthermore, and its derivative satisfy the recurrence relations

(3)            (4)

Proof

(1)

By multiply both sides with respect to , we obtain

By derivative both sides with respect to , we obtain

By divide both sides with respect to , we obtain

(2)

By multiply both sides with respect to , we obtain

By derivative both sides with respect to , we obtain

By multiply both sides with respect to , we obtain

(3) Subtracting from

Example 2  Application of Theorem 3 in Evaluation and Integration

so .

When , then .

When , then

When , then .

When , then .

Example 3  Elementary for Half-Integer Order

,

Proof

Using Gamma function property

Using Gamma function property

Remark  ,

Example  Further Elementary Bessel Functions

so

When , we obtain

so

When , we obtain

*5.5 Bessel Functions of the . General Solution

: Bessel Equation of the Second Kind

When , Bessel’s function is

(1)        .

Then the indicial equation has a double root .

We first solution have only one solution, .

We see that the desired solution must be of the form

(2)        .

We substitute and its derivatives

into . Then the sum of the three logarithmic terms , and is zero because is a solution of .

For this gives

,   thus

For this yields

,   thus

and in general

(3)

Using the short notations

(4)

and inserting and into , we obtain the result

Since and are linearly independent functions, they form a basis of for .

It is customary to choose and , where the number is the so-called Euler constant, which is defined as the limit of

as approaches infinity. The standard particular solution thus obtained is called the Bessel function of the second kind of oder zero or Neumann’s function of oder zero and is denoted by .

(6)

For small the function behaves about like , and .

Bessel Equations of the Second Kind

For this reason we introduce a standard second solution defined for all by the formula

(7)   (a)

(b)

This function is called the Bessel function of the second kind of oder or Neumann’s function of oder . Figure 109 shows and .

The result is

(8)

where , and , , ,

For the last sum in is to be replaced by .

Furthermore, it can be shown that

.

Sage Coding

 # (주의) bessel 관련 아래 코드는 https://sagecell.sagemath.org/  에서 실행하세요. f = bessel_Y(0, x)      # Bessel 함수 정의 g = bessel_Y(1, x) h = bessel_Y(2, x) plot(f, (x, 0, 10), ymin = -0.5, ymax = 0.5) + plot(g, (x, 0, 10), color = 'red') + plot(h, (x, 0, 10), color = 'green')

Evaluate

Theorem 1  General solution of Bessel’s Equation

A general solution of Bessel’s equation for all values of (and ) is

(9)        .

For this purpose the solutions

are frequently used. These linearly independent functions are called Bessel function of the third kind of oder or first and second Hankel function of oder .

[한빛 아카데미] Engineering Mathematics with Sage:

[저자] 이상 구, 김영 록, 박준 현, 김응 기, 이재 화

Contents

A. 공학수학 1 – 선형대수, 상미분방정식+ Lab

Chapter 01 벡터와 선형대수 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-1.html

Chapter 02 미분방정식의 이해 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-2.html

Chapter 03 1계 상미분방정식 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-3.html

Chapter 04 2계 상미분방정식 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-4.html

Chapter 05 고계 상미분방정식 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-5.html

Chapter 06 연립미분방정식, 비선형미분방정식 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-6.html

Chapter 07 상미분방정식의 급수해법, 특수함수 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-7.html

Chapter 08 라플라스 변환 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-8.html

B. 공학수학 2 - 벡터미적분, 복소해석 + Lab

Chapter 09 벡터미분, 기울기, 발산, 회전 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-9.html

Chapter 10 벡터적분, 적분정리 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-10.html

Chapter 11 푸리에 급수, 적분 및 변환 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-11.html

Chapter 12 편미분방정식 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-12.html

Chapter 13 복소수와 복소함수, 복소미분 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-13.html

Chapter 14 복소적분 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-14.html

Chapter 15 급수, 유수 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-15.html

Chapter 16 등각사상 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-16.html

Made by Prof. Sang-Gu LEE  sglee at skku.edu

http://matrix.skku.ac.kr/sglee/   with Dr. Jae Hwa LEE