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: E12 (금 12:00 - 14:45), 반도체관 400126호 담당교수: 김응기 박사 
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주차 |
주교재 |
부교재 |
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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 |
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2 |
11.7: Fourier Integral 11.8: Fourier Cosine and Sine Transforms |
11.4 푸리에 급수의 수렴 11.5 스투룹-리우빌 정리와 직교함수 |
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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 |
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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 |
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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 |
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6 |
8.1: Eigenvalues and Eigenvectors 8.4: Eigenbases, Diagonalization , Quadratic Forms |
1.6 고유값과 고유벡터 1.7 닮음, 행렬의 대각화, 이차형식 |
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7 |
4.1: System of ODEs |
6.1 연립미분방정식이란 6.2 동차 선형연립미분방정식 6.4 기본행렬과 지수행렬을 이용한 연립미분방정식의 풀이 6.6 연립미분방정식의 분리 |
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8 |
Review and Project proposal (presentation) Midterm Exam |
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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 |
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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 |
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11 |
(newly added) Conditional Probability and Independence, Bayes' theorem 24.5: Random Variables. Probability Distributions 24.6: Mean and Variance of a Distribution |
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12 |
24.7: Binomial, Poisson, and Hypergeometric Distributions |
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13 |
24.8: Standard deviation and Normal Distribution 24.9: Distributions of Several Random Variables |
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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 |
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15 |
Review and Project / Homework presentation |
복습, 프로젝트(과제) 발표 |
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web |
http://matrix.skku.ac.kr/2010-BSM-S/EngMath-2/engineering_math.html (공학수학 2 동영상강의) |
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16 |
Final Exam |
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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.
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Definition |
Periodic Functions |
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Periodic function is a function that repeats its values in regular intervals or periods. (1) |
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, 
is a 
is a 
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 
.
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Remark |
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, 
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 
.
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Definition |
Fourier series |
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Let 1. 2. 3. |
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satisfy the following conditions
is defined in the interval 
.
and 
are sectionally continuous in the interval 
.
i.e. 
is periodic with period 
.|
Theorem |
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Fourier series of (5) with the Fourier coefficients Fourier coefficients (6) (a) (b) (c) |
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is
.
and 
.
and 
are
Proof
(a) 
is integration on 
,
.
is odd function. 
(b) Multiplying 
by 
and integration on 
,
.
(c) Multiplying 
by 
and integration on 
,
. 
.
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Example 1 |
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Find the Fourier coefficients of the periodic function (7)
Fig 260 Given function |
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in Fig. 260.
and 
.
(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/
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 
,
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 
.
having continuous first and second derivatives. Integrating 
by parts, we obtain
.
. Since 
is continuous in the integral of integration, we have
. Furthermore 
.
.
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
with coefficients 
represents 
are substantially more complicated:
Left-hand and right-hand limits
of the function 
at 
is defined as the limit of 
as 
approaches 
from the left and is commonly denoted by 
. Thus
as 
through positive values.
and
as 
through positive values.
at 
are defined as the limit of
and 
,
through positive values.
is continuous at 
, the last term in both numerators is simply 
.
.
and its right-hand limit there is 
.
. The Fourier series 
of the wave does indeed converge to this value when 
because then all its terms are 
.
.
to any period 
, for the function 
, simply by a
-axis.
is even (“Fourier cosine series”).
is odd (“Fourier sine series”).
given for 
in two Fourier series, one having only cosine
to Any Period 
to be period 
is effected by a suitable change of scale, as follows.
have period 
.
such that 
, as a function of 
, has period 
. If we set
where 
corresponds to 
.
is a function of 
, 
has period 
. A Fourier series is
simplifies calculations. Since
we have 
from 
to 
.
of period 
the Fourier series
of period 
is
given by the 
to 
or over any other interval of length 
.
is integration on 
, we have
.
by 
and integration on 
, 
.
by 
and integration on 
, 
.
of period 
is
.
From 
.
From 
is even 
is odd
From 
, 
.
.
,
.
From
.
From
for all
,
.
From
and thus 
.
is even 
is odd 
,
,
,
,
,
.
.
, where
.
From
From
,
is odd 
is even 
From
.
for all
,
,
,
,
,
,
,
.
and
.
.
is 
is even function 
.
.
is 
is odd function 
.
is a “
(
is even function)
for all 
.
is integration from on 
, 
. 
.
by 
and integration on 
,
.
(
for all
.
.
(
for all
is integration
.
.
(
for all
.
.
.
.
)
).
).
if 
and
.
. Sawtooth wave
and
.
is odd function
,
,
,
,
.
.
.
,
,
,
.
is given only on half the range, half the interval of periodicity of length 
, 
, 
for 
, 
, 
for 
continued as an even periodic function of period 
continued as an odd periodic function of period 
.
.
.
.
, 
, 
,
, Hence the first half-range expansion of
.
, of period
.
.
.
.
.
of our function are
, 
, 
, 
, 
, 
for all even 
, Hence the other half-range expansion of 
is
.
, of period 
.
