2019학년도 2학기

                   반도체 공학과 공학수학2


   주교재: Erwin Kreyszig, Engineering Mathematics, 10th Edition

   부교재: 이상구 외 4인, 최신공학수학 Ⅱ,  1st Edition  

   실습실: http://www.hanbit.co.kr/EM/sage/

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

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

   강의시간:  공학수학2, (화16:30-17:45)   (목15:00-16:15)

   담당교수:  김응기 박사



Week 10


10주차

13.6: Trigonometric and Hyperbolic Functions. Euler’s Formula

  13.5 복소 삼각함수와 복소 쌍곡함수


13.7: Logarithm. General Power. Principal Value

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



13.6. Trigonometric and Hyperbolic Functions

Euler’s Formula


Euler formulas

    ,  

    .

     :

     : .


Definitions for complex value :

(1)     ,          .

(2)     ,                 .

(3)     ,                 .

Formulas for the derivatives

(4)                   

                 .

       

               

       

       

               


Equation also shows that Euler’s formula is valid in complex:

(5)         for all .


Example 1  Real and Imaginary Parts. Absolute value. Periodicity

Show that

(6)    

and

(7)    

and give some applications of these formulas.

Solution

From ,

(8)     ,   

    

         

         

         

    

         

         

         

From and , , we obtain

    

           .

    

          .


Example 2  Solutions of Equations. Zeros of and

Solve .

Solution

(a)             

     :   (by multiplication by )

This is a quadratic equation in ,

     and .

    

Thus or .

                

                

            where .

     or where .

(b)       , .

             

          .

Hence        

           by multiplication

      

             

(c)       , .

              

          .

Hence   

    

     by multiplication

             

              .

Here the only zero of and are those the real cosine and sine function.



General formulas


Addition rules

(9)    

and the formula

(10)    .

    

               

               

    

              

              

    


Hyperbolic Functions

The complex hyperbolic cosine and sine are defined by the formula

(11)     ,       .

(12)     ,           .

        ,           .

Formulas for the derivatives

(13)     ,          

                    

             .


Complex Trigonometric and Hyperbolic Functions Are Related.

If in , we replace by and then use , we obtain

(14)     ,   

         .

Similarly, if in we replace by and then use , we obtain conversely

(15)     ,

        

              



13.7. Logarithm. General Power. Principal Value


(or ) is natural logarithm of .

 is the inverse of the complex exponential function; that is,

     ()       .

      

    inverse of the complex exponential function


If we set and , this becomes

    .

           

    

         .

            .

    ,  ,  

   ,     .

          ,

where is the familiar real natural logarithm of the positive number .


Hence is given by

(1)         .


       ,  

the complex natural logarithm is infinitely many-valued.


     


    

    .


    

      

    


The value of corresponding to the principal value is denoted by .

 is the principal value of .

(2)       .

       is the principal value of


The uniqueness of for given implies that is single-valued. Since the other values of differ by integer multiples of , the other values of are given by

    

              

(3)       .

The same real part,

The imaginary parts differ by the integer multiples of .


 is positive real, then .

          

 is negative real, then and

     ( negative real).


From and for positive real we obtain

(4a)   

as expected, but since is multivalued, so is

(4b)    ,  .


Example 1  Natural Logarithm. Principle Value

                            

                                

                       

                        

                             

                            

                         

                           

              

           그림입니다.

  Fig. 337  Some values of in Example 1


The familiar relations for the natural logarithm continue to hold for complex value, that is,

(5)     ,   

but these relations are to be understood in the sense that each value of one side is also contained among the values of the other side.


Sage Coding

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


Example 2  Illustration of the Functional Relation in Complex

Let

    .

If we take the principal values

    ,

then holds provided we write ; however, it is not true for the principal value, .


    ,

    

then

            

    .


    


Theorem 1  Analyticity of the Logarithm

For every formula defines a function, which is analytic, except at and the negative real axis, and has the derivative

(6)        ( not or negative real)

Proof

We show that the Cauchy-Riemann equations are satisfied. From we have

    

where the constant is a multiple of . By differentiation,

    

    

    

 is analytic.


Hence the Cauchy-Riemann equations hold. Formula in Sec. 13.4 now gives ,

    

          .


Each of the infinitely many functions in is called a branch of the logarithm. The negative real axis is known as a branch cut and is usually graphed as show in Fig. 338. The branch for is called the principal branch of .

       

     Fig.. Branch cut for



General Powers

General powers of a complex number are defined by the formula

(7)     ( complex, ).

Since is infinitely many-valued, will, in general, be multivalued. The particular value

       

is called the principal value of .

If , then is single-valued and identical with the usual power of . If , the situation is similar.

If , where , then

         ,

the exponent is determined up to multiples of and we obtained then distinct values of the root, in agreement with the result in Sec. 13.2.

If , the quotient of two positive integers, the situation is similar, and has only finitely many distinct values. However, if is real irrational or genuinely complex, then is infinitely many-valued. 


Example 3  General Power

    . ()

All these values are real, and the principal value is .

Similarly, by direct calculation and multiplying out in the exponent,

                  

             

             

             

             

             

             

             

              ( is integer).

From we see that for any complex number ,

(8)     .



http://matrix.skku.ac.kr/sglee/ 


[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


Copyright @ 2019 SKKU Matrix Lab. All rights reserved.
Made by Manager: Prof. Sang-Gu Lee and Dr. Jae Hwa Lee
*This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2017R1D1A1B03035865).