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    

 

주차

주교재

부교재

1

11.1: Fourier Series

11.2: Arbitrary Period. Even and Odd Functions. Half-Range Expansions

11.1 푸리에 급수란

11.2 푸리에 급수와 경계조건

11.3 푸리에 급수의 기하학적 접근

web

http://www.hanbit.co.kr/EM/sage/2_chap11.html 

2

11.7: Fourier Integral

11.8: Fourier Cosine and Sine Transforms

11.4 푸리에 급수의 수렴

11.5 스투룹-리우빌 정리와 직교함수

web

http://www.hanbit.co.kr/EM/sage/2_chap11.html 

3

11.9: Fourier Transform. Discrete and Fast Fourier Transforms

12.1: Basic Concepts of PDEs

12.2: Modeling: Vibrating String, Wave Equation

11.6 푸리에 변환

12.1 편미분방정식이란

web

http://www.hanbit.co.kr/EM/sage/2_chap12.html 

4

(briefly)

12.3: Solution by Separating Variables. Use of Fourier Series

12.6: Heat Equation: Solution by Fourier Series. Steady Two-Dimensional Heat Problems. Dirichlet Problem

12.7: Heat Equation: Solution by Fourier Integrals and Transforms

12.3 파동방정식

12.4 확산방정식

web

http://www.hanbit.co.kr/EM/sage/2_chap12.html 

5

(briefly Review)

7.2: Matrix Multiplication

7.3: Linear Systems of Equations

7.4: Linear Independence, Rank of Matrix, Vector Space

7.5: Solutions of Linear System

7.6: Second and Third-Order Determinants

7.7: Determinants.

7.8: Inverse of Matrix

1.1 행렬의 성질과 연산

1.2 선형연립방정식

1.3 일차독립과 일차종속, 계수

1.4 행렬식과 여인자 전개

1.5 역행렬과 크래머 법칙

web

http://www.hanbit.co.kr/EM/sage/1_chap1.html 

6

8.1: Eigenvalues and Eigenvectors

8.4: Eigenbases, Diagonalization , Quadratic Forms

1.6 고유값과 고유벡터

1.7 닮음, 행렬의 대각화, 이차형식

web

http://www.hanbit.co.kr/EM/sage/1_chap1.html 

7

4.1: System of ODEs

6.1 연립미분방정식이란

6.2 동차 선형연립미분방정식

6.4 기본행렬과 지수행렬을 이용한 연립미분방정식의 풀이

6.6 연립미분방정식의 분리

web

http://www.hanbit.co.kr/EM/sage/1_chap6.html

 

8

Review and Project proposal (presentation)

Midterm Exam

9

13.1: Complex Numbers. Complex Plane

13.2: Polar Form of Complex Numbers. Powers and Roots

13.3: Derivative. Analytic Function

13.1 복소수

13.2 복소함수

web

http://www.hanbit.co.kr/EM/sage/2_chap13.html

10

(briefly)

13.4: Cauchy–Riemann Equations. Laplace’s Equation

13.5: Exponential Function

13.6.: Trigonometric and Hyperbolic Functions. Euler’s Formula

13.7: Logarithm. General Power. Principal Value

13.3 코시-리만방정식

13.4 복소 지수함수와 복소 로그함수

13.5 복소 삼각함수와 복소 쌍곡함수

web

http://www.hanbit.co.kr/EM/sage/2_chap13.html 

11

(newly added) Conditional Probability and Independence, Bayes' theorem

24.5: Random Variables. Probability Distributions

24.6: Mean and Variance of a Distribution

https://youtu.be/ACWuV16tdhY 

https://youtu.be/Hj1pgap4UOY 

https://youtu.be/JbYtinw-CqI 

web

http://matrix.skku.ac.kr/2018-album/R-Sage-Stat-Lab-1.html 

12

24.7: Binomial, Poisson, and Hypergeometric Distributions

https://youtu.be/uzkc-qNVoOk 

https://youtu.be/iG995W0XefU 

https://youtu.be/peEsXbdMY_4 

web

http://matrix.skku.ac.kr/2018-album/R-Sage-Stat-Lab-2.html 

13

24.8: Standard deviation and Normal Distribution

24.9: Distributions of Several Random Variables

https://youtu.be/0ZstEh_8bYc 

https://youtu.be/JNm3M9cqWyc 

web

http://matrix.skku.ac.kr/2018-album/R-Sage-Stat-Lab-2.html 

14

(if time permits)

16.1: Laurent Series

16.2: Singularities and Zeros. Infinity

16.3: Residue Integration Method

16.4: Residue Integration of Real Integrals

15.1 복소 수열과 복소 무한급수

15.2 테일러 급수와 로랑급수

web

http://www.hanbit.co.kr/EM/sage/2_chap15.html 

15

Review and Project / Homework presentation

복습, 프로젝트(과제) 발표

web

http://matrix.skku.ac.kr/2010-BSM-S/EngMath-2/engineering_math.html (공학수학 2 동영상강의)

16

Final Exam

 

 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

(a) is integration on ,

   

              

             

         .

      

       is odd function.     

(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

Evaluate

  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


Derivation of the Euler Formulas (6)


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 .

 


Example 2

Convergence at a Jump as Indicated in Theorem 2

The rectangular wave in Example 1 has a jump at .

Its left-hand limit there is and its right-hand limit there is .

Hence the average of these limits is . The Fourier series of the wave does indeed converge to this value when because then all its terms are .

Similarly for the other jumps. This is in agreement with Theorem .


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


The transition from period to be period is effected by a suitable change of  scale, as follows.

 have period .

A new variable such that , as a function of , has period . If we set

(1)            where

then corresponds to .

 is a function of , has period . A Fourier series is

(2)              

with coefficients obtained from (6) in the last section

(3)     

        

        

We could use these formulas directly, but the change to simplifies calculations. Since

(4)                we have 

and we integrate over from to .

Consequently, we obtain for a function of period the Fourier series


Theorem

Fourier series of the function of period is

(5)                        

with the Fourier coefficients of given by the Euler formulas

              (a)  

(6)           (b)  

              (c)  

And we can integrate from to or over any other interval of length .

Proof

(a) is integration on , we have

    

                

                

          .

(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

Evaluate

  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

Evaluate

  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

Evaluate

  -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 .

 

Definition

Odd Functions

       is odd function.

Graph of odd function is symmetric about the original point.

 is odd 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 .

Proof

 is integration from on

    

               

               

               

               

               

      .      .

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

 is integration on

    

               

               

               

               

               

           .


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

Evaluate

  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

 

Evaluate

  -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

Evaluate

  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