2018학년도 2학기

   반도체 공학과 공학수학2 (GEDB005) 강의교안  (2018학년도용)

       주교재: Erwin Kreyszig, Engineering Mathematics, 10th Edition

       부교재: 최신공학수학 I 과 II, 한빛출판사 및 (영문) 다변량 Calculus (이상구, 김응기 etal)   (http://www.hanbit.co.kr/EM/sage/)

       강의시간: BD1615(화 16:30-17:45/ 목 15:00-16:15), 반도체관 400102호 담당교수: 김응기 박사    

 Week 1

Chapter 11. Fourier series

11.1. Fourier series

Fourier series is an expansion of a periodic function in terms of an infinite sum of cosines and sines.

Definition	Periodic Functions
Periodic function is a function that repeats its values in regular intervals or periods. (1)                   ,       is a period of                                  is a periodic function

                                       Periodic function of period

The smallest positive period is the fundamental period.

For example, cosine, sine, tangent and cotangent are periodic function.

                   periodic

Not periodic : , , , , and

(2)       ,  ,   

                               has periodic

 and have periodic       has the periodic with any constants and .

Remark
,       periodic ,       periodic               periodic

For instance,

(3)              

All these functions have the period from the so-called trigonometric system.

Trigonometric system

Cosine and sine functions having the period

Trigonometric series has a series of the form

(4)      

 are constants and the coefficients of the series.

Each term has the period .

If the coefficients are such that the series converges, its sum will be a function of period .

Definition	Fourier series
Let satisfy the following conditions 1. is defined in the interval . 2. and are sectionally continuous in the interval . 3. i.e. is periodic with period .

Theorem
Fourier series of is (5)                        . with the Fourier coefficients and . Fourier coefficients and are (6)           (a)              (b)              (c)

Proof

(b) Multiplying by and integration on ,

(c) Multiplying by and integration on ,

Example 1
Find the Fourier coefficients of the periodic function in Fig. 260.  (7)                 and  . Fig 260 Given function (Periodic rectangular wave)

Solution

      From

 .

      From

    for all

      From

  .

                            .

Fourier coefficients of given function are

                                .

 .

Fourier series of is

(8)                       

                               .

Partial sums are

    ,    

    ,

      etc.

Given series is convergent.

Given  series has the sum .

 is discontinuous at and .

All partial sum have the value zero at and .

the arithmetic mean of the limit and of our function at and .

 is the sum of the series at

                                                  .

The values of various series with constant terms can be obtained by evaluating Fourier series at specific points.

                      Fig 261. First three partial sums of the corresponding Fourier series

Sage Coding

http://math3.skku.ac.kr/  http://sage.skku.edu/  http://mathlab.knou.ac.kr:8080/

f1(x) = -2 f2(x) = 2 f = Piecewise([[(-pi,0),f1],[(0,pi),f2]]) p1 = plot(f) fps = f.fourier_series_partial_sum(10, pi) print fps p2 = plot(fps, (x, -pi,pi), color='red') p1 + p2

  8/3*sin(3*x)/pi + 8/5*sin(5*x)/pi + 8/7*sin(7*x)/pi + 8/9*sin(9*x)/pi + 8*sin(x)/pi

※ Caution : http://aleph.sagemath.org/  http://sagecell.sagemath.org/

   Command change required when using

f = piecewise([((-pi,0),-2),((0,pi),2)]) p1 = plot(f, (x, -pi, pi)) fps = f.fourier_series_partial_sum(10) print fps p2 = plot(fps, (x, -pi, pi), color='red') p1 + p2

Theorem 1	Orthogonality of the Trigonometric System (3)
The trigonometric system is orthogonal on the interval ; that is, the integral of the product of any two function in over that interval is , so that for any integral and ,                           (a)  (9)                       (b)                            (c)

Proof Skip

Convergence and Sum of a Fourier Series

Theorem 2	Representation by a Fourier Series
is periodic with period . is piecewise continuous in the interval . have the a left-hand derivative and a right-hand derivative at each point of that interval. Then the Fourier series of converges. Its sum is , except at points where is discontinuous.  There the sum of the series is the average of the left-hand and right-hand limit of at .

Proof

We prove convergence for a continuous function having continuous first and second derivatives. Integrating by parts, we obtain

The first term on the right is zero. Another integration by parts gives

                                 .

The first term on the right is zero because of the periodicity and continuity of . Since is continuous in the integral of integration, we have

for an appropriate constant . Furthermore .

It follows that

                           .

Similarly, for all . Hence the absolute value of each term of the Fourier series of is at most equal to the corresponding term of the series

which is convergent.

Hence, that Fourier series converges and the proof is complete.

The proof of convergence in the case of a piecewise continuous function

   and the proof under the assumptions in the theorem the Fourier series with coefficients represents are substantially more complicated:

Proof of theorem2 Left-hand and right-hand limits

 of the function

                        Left- and right-hand limits

Left-hand limit of at is defined as the limit of as approaches from the left and is commonly denoted by . Thus

                                             as through positive values.

Right-hand limit is denoted by and

                                     as through positive values.

Left-hand limit and right-hand limit of at are defined as the limit of

                                 and  ,

respectively, as through positive values.

Of course if is continuous at , the last term in both numerators is simply .

11.2 Arbitrary Period. Even and Odd Functions.

Half-Range Expansions

Orientation

This section concerns three topics:

1. Transition from period to any period , for the function , simply by a

   transformation of scale on the -axis.

2. Simplifications. Only cosine terms if is even (“Fourier cosine series”).

  Only sine terms if is odd (“Fourier sine series”).

3. Expansion of given for in two Fourier series, one having only cosine

terms and the other only sine terms (“half-range expansions”).

1. From Period to Any Period

(b) Multiplying by and integration on , 

(c) Multiplying by and integration on , 

Example 1	Periodic Rectangular Wave
Find the Fourier series of the function of period   is .

Solution

      From

 .

      From

  is even     

  is odd

      From

 , .

Fourier series is

                .

Sage Coding

http://math3.skku.ac.kr/  http://sage.skku.edu/  http://mathlab.knou.ac.kr:8080/

f1(x) = 0 f2(x) = 2 f3(x) = 0 f = Piecewise([[(-2,-1),f1],[(-1,1),f2],[(1,2),f3]]) p1 = plot(f) fps = f.fourier_series_partial_sum(10, 2) print fps p2 = plot(fps, (x, -2,2), color='red') p1 + p2

  4*cos(1/2*pi*x)/pi - 4/3*cos(3/2*pi*x)/pi + 4/5*cos(5/2*pi*x)/pi -

4/7*cos(7/2*pi*x)/pi + 4/9*cos(9/2*pi*x)/pi + 1

※ Caution : http://aleph.sagemath.org/  http://sagecell.sagemath.org/

  사용 시 명령어 변경 필요함

f = piecewise([((-2,-1), 0), ((-1,1), 2), ((1,2), 0)]) p1 = plot(f, (x, -2, 2)) fps = f.fourier_series_partial_sum(10) print fps p2 = plot(fps, (x, -2, 2), color='red') p1 + p2

Example 2	Periodic Rectangular Wave. Change of Scale
Find the Fourier series of the function of period , is .

Solution

      From

 .

      From

    for all

, .

      From

                        and thus     .

   is even     

   is odd     

Fourier coefficients are

                                , , , , , .

Fourier series of is

                           .

Sage Coding

 http://math3.skku.ac.kr/  http://sage.skku.edu/  http://mathlab.knou.ac.kr:8080/

f1(x) = -2 f2(x) = 2 f = Piecewise([[(-2,0),f1],[(0,2),f2]]) p1 = plot(f) fps = f.fourier_series_partial_sum(10, 2) print fps p2 = plot(fps, (x, -2, 2), color='red') p1 + p2

  8*sin(1/2*pi*x)/pi + 8/3*sin(3/2*pi*x)/pi + 8/5*sin(5/2*pi*x)/pi +

8/7*sin(7/2*pi*x)/pi + 8/9*sin(9/2*pi*x)/pi

※ Caution : http://aleph.sagemath.org/  http://sagecell.sagemath.org/

            사용 시 명령어 변경 필요함

f = piecewise([((-2,0), -2), ((0,2), 2)]) p1 = plot(f, (x, -2, 2)) fps = f.fourier_series_partial_sum(10) print fps p2 = plot(fps, (x, -2, 2), color='red') p1 + p2

Example 3	Half-Wave Rectifier
A sinusoidal voltage , where is time, is passed through a half-wave rectifier that clips the negative portion of the wave . Half-wave rectifier

Solution

      From

      From

,

 is odd     

 is even     

      From

   .

     for all

Fourier coefficients are

                        , , , , , , , .

Fourier coefficients are

                        and for all .

Fourier series of is

                   .

Sage Coding

http://math3.skku.ac.kr/  http://sage.skku.edu/  http://mathlab.knou.ac.kr:8080/

w = 3 E = 5 L = pi/w f1(x) = 0 f2(x) = E*sin(w*x) f = Piecewise([[(-L,0),f1],[(0,L),f2]]) p1 = plot(f) fps = f.fourier_series_partial_sum(6, L) print fps p2 = plot(fps, (x, -L,L), color='red') p1 + p2

  -10/3*cos(6*x)/pi - 2/3*cos(12*x)/pi + 5/pi + 5/2*sin(3*x)

※ Caution : http://aleph.sagemath.org/  http://sagecell.sagemath.org/

사용 시 명령어 변경 필요함

w = 3 E = 5 L = pi/w f = piecewise([((-L,0), 0), ((0,L), E*sin(w*x))]) p1 = plot(f, (x, -L, L)) fps = f.fourier_series_partial_sum(6) print fps p2 = plot(fps, (x, -L, L), color='red') p1 + p2

2. Simplifications: Even and Odd Functions

Definition	Even Functions
is even function. Graph of even function is symmetric about the vertical axis.  is even function      . For example .

Fourier Cosine Series, Fourier Sine Series

Theorem	Fourier Cosine Series of Period 2L
Fourier series of an even function of period is a “Fourier Cosine Series” (1)                           ( is even function) with coefficients (2)                                         for all .

Multiplying by and integration on ,

Theorem	Fourier Sine Series of Period 2L
Fourier series of an odd function of period is a “Fourier Sine Series” (3)                             ( is odd function) with coefficients (4)                     for all .

Proof

Multiplying by and integration on , 

          .

Case of Period .

Theorem	Fourier Cosine Series of Period
Fourier series of an even function of period is a “Fourier Cosine Series” (1)                           ( is even function) with coefficients (2)                                         for all .

Proof

Multiplying by and integration on , 

          .

Theorem	Fourier Sine Series of Period
Fourier series of an odd function of period is a “Fourier Sine Series” (3)                             ( is odd function) with coefficients (4)                     for all .

Proof

Multiplying by and integration on , 

            .

Theorem 1	Sum and Scalar Multiple
Fourier coefficients of a sum are the sums of the corresponding Fourier coefficients of and . Fourier coefficients of are times the corresponding Fourier coefficients of .

Fourier coefficients of (Fourier coefficients of ) (Fourier coefficients of ).

Fourier coefficients of (Fourier coefficients of ).

Example 5	Sawtooth Wave
Find the Fourier series of the function of period is    if      and . Fig 268 The function . Sawtooth wave

Solution

       and .

     is odd function

Fourier coefficients of function are

            , , , , .

      .

Fourier series of is

Fourier series of is

Fourier series of is

                        .

                           Fig 269 partial Sums , , , in Example 5

Sage Coding

http://math3.skku.ac.kr/  http://sage.skku.edu/  http://mathlab.knou.ac.kr:8080/

f1(x) = x f2(x) = pi f = Piecewise([[(-pi,pi),f1]]) p1 = plot(f1+f2,(x,-pi,pi)) fps = f.fourier_series_partial_sum(10,pi) print fps + f2(x) p2 = plot(fps+f2, (x,-pi,pi), color='red') p1 + p2

  pi - sin(2*x) + 2/3*sin(3*x) - 1/2*sin(4*x) + 2/5*sin(5*x) - 1/3*sin(6*x) + 2/7*sin(7*x) - 1/4*sin(8*x) + 2/9*sin(9*x) + 2*sin(x)

※ Caution : http://aleph.sagemath.org/  http://sagecell.sagemath.org/

사용 시 명령어 변경 필요함

f = piecewise([((-pi,pi), x)]) p1 = plot(f+pi,(x,-pi,pi)) fps = f.fourier_series_partial_sum(10) print fps + pi p2 = plot(fps+pi, (x,-pi,pi), color='red') p1 + p2

3. Half-Range Expansions

Half-range expansions are Fourier series.

Even periodic extension and odd periodic extension.

Both extensions have period .

This motivates half-range expansions : is given only on half the range, half the interval of periodicity of length

Definition	Even periodic extension
, , for half-range cosine series.

Definition	Odd periodic extension
,  ,  for half-range sine series.

Definition	Half-Range Expansions
The given function

Definition	Even Periodic Extension
continued as an even periodic function of period

Definition	Odd Periodic Extension
continued as an odd periodic function of period

Example 6	“Triangle” and Its Half-Range Expansions
Find the two half-range expansions of the function Fig 271 The given function in Example 6

Solution

(a) Even periodic extension.

From we obtain

         .

For the integer we obtain by integration by parts

                        .

                            .

                            .

Fourier coefficients of our function are

                ,  ,  ,

Since the if , Hence the first half-range expansion of is

                   .

This Fourier cosine represents the even periodic extension of the given function , of period .

                                Even extension

Sage Coding

http://math3.skku.ac.kr/  http://sage.skku.edu/  http://mathlab.knou.ac.kr:8080/

L = 6 K = 3 f1(x) = 2*K/L*x f2(x) = 2*K/L*(L-x) f = Piecewise([[(-L,-L/2),f2(-x)],[(-L/2,0),f1(-x)],[(0,L/2),f1],[(L/2,L),f2]]) p1 = plot(f) fps = f.fourier_series_partial_sum(10,L) print fps p2 = plot(fps, (x,-L,L), color='red') p1 + p2

  -12*cos(1/3*pi*x)/pi^2 - 4/3*cos(pi*x)/pi^2 + 3/2

※ Caution : http://aleph.sagemath.org/  http://sagecell.sagemath.org/

사용 시 명령어 변경 필요함

L = 6 K = 3 f1(x) = 2*K/L*x f2(x) = 2*K/L*(L-x) f = piecewise([((-L,-L/2),f2(-x)),((-L/2,0),f1(-x)),((0,L/2),f1),((L/2,L),f2)]) p1 = plot(f, (x,-L,L)) fps = f.fourier_series_partial_sum(10) print fps p2 = plot(fps, (x,-L,L), color='red') p1 + p2

(b) Odd periodic extension.

         .

For the integer we obtain by integration by parts

                     .

                          .

                          .

Hence the Fourier coefficients of our function are

              ,   ,   ,   ,   ,  

Since the for all even , Hence the other half-range expansion of is

       .

This Fourier sine represents the odd periodic extension of the given function , of period .

                            Odd extension

Sage Coding

http://math3.skku.ac.kr/  http://sage.skku.edu/  http://mathlab.knou.ac.kr:8080/

L = 6 K = 3 f1(x) = 2*K/L*x f2(x) = 2*K/L*(L-x) f = Piecewise([[(-L,-L/2),-f2(-x)],[(-L/2,0),-f1(-x)],[(0,L/2),f1],[(L/2,L),f2]]) p1 = plot(f) fps = f.fourier_series_partial_sum(10,L) print fps p2 = plot(fps, (x,-L,L), color='red') p1 + p2

  24*sin(1/6*pi*x)/pi^2 - 8/3*sin(1/2*pi*x)/pi^2 + 24/25*sin(5/6*pi*x)/pi^2

  - 24/49*sin(7/6*pi*x)/pi^2 +8/27*sin(3/2*pi*x)/pi^2

※ Caution : http://aleph.sagemath.org/  http://sagecell.sagemath.org/

사용 시 명령어 변경 필요함

L = 6 K = 3 f1(x) = 2*K/L*x f2(x) = 2*K/L*(L-x) f = piecewise([((-L,-L/2),-f2(-x)),((-L/2,0),-f1(-x)),((0,L/2),f1),((L/2,L),f2)]) p1 = plot(f, (x,-L,L)) fps = f.fourier_series_partial_sum(10) print fps p2 = plot(fps, (x,-L,L), color='red') p1 + p2

[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 Jae-Hwa LEE