Chapter 2 Complex numbers (복소수) Mathematical methods in the physical sciences 3rd edition Mary L. Boas Chapter 2 Complex numbers (복소수) Lecture 4 Introduction of complex numbers 고등수학 10-가 2장 실수와 복소수에 나옴 1

1. Introduction ex. 2

(READING) Once the new kind of number is admitted into our number system, fascinating possibilities open up. Can we attach any meaning to marks like sin i, e^i, ln (1+i)? We’ll see later that we can and that, in fact, such expressions may turn up in problems in physics, chemistry, and engineering, as well as, mathematics. When people first considered taking square roots of negative numbers, they felt very uneasy about the problem. They thought that such numbers could not have any meaning or any connection with reality (hence the term “imaginary”). They certainly would not have believed that the new numbers could be of any practical use. Yet complex numbers are of good importance in a variety of applied fields; for example, the electrical engineer would, to say the least, be severely handicapped without them. The complex notation often simplifies setting up and solving vibration problems in either dynamical or electrical systems, and is useful in solving many differential equations which arise from problems in various branches of physics. 3

2. Real and imaginary parts of a complex number (복소수의 실수와 허수 부분) (복소수의 실수와 허수 부분) y: imaginary part (not imaginary!!) x: real part 4

- Complex plane: similar to the xy plane 5

Rectangular form (x,y) vs. Polar form (r,) (직교형태 VS 극좌표 형태) 6

Example) cf.  : radian 7

4. Terminology and notation ex) principal angle 8

- Complex conjugate (켤레 복소수) 9

5. Complex algebra (복소수 연산) A. Simplifying to x+iy form squaring 10

B. Complex conjugate note) We can get the conjugate of an expression containing i’s by just changing the signs of all the i terms. C. Absolute value 11

D. Complex equations 12

E. Graphs ex.1 ex2. ex3. ex.4 13

Chapter 2 Complex numbers Mathematical methods in the physical sciences 3rd edition Mary L. Boas Chapter 2 Complex numbers Lecture 5 Euler formula & roots and powers

6. Complex infinite series (복소수 무한 급수) In this case, we call the complex series convergent. 7. Complex power series; Disk of convergence (복소수 멱급수 ; 수렴 원판) ex. cf. real vs. complex

2.8 Elementary function of complex numbers (복소수 기본함수) - elementary functions: powers, roots, trigonometric, inverse trigonometric, logarithmic, exponential, and combinations of these. - Elementary functions of complex numbers behave just like those of real numbers

2.9 Euler’s formula

Ex. Find the graph expressing a given z.

- Multiplication, division ex.

10. Powers and roots of complex numbers (복소수의 멱수와 근)

ex.2 Cube roots of 8?

ex.3 Find the plot all values of

ex. 4

Chapter 2 Complex numbers Mathematical methods in the physical sciences 3rd edition Mary L. Boas Chapter 2 Complex numbers Lecture 6 application

11. The exponential and trigonometric functions (지수함수와 삼각함수) - exponential function - trigonometric function

12. Hyperbolic functions (쌍곡함수) - The other hyperbolic functions are named and defined in a similar way to parallel the trigonometric functions:

13. Logarithms (로그함수) - Since  has an infinite values (all differing by multiples of 2), a complex number has infinitely many logarithms. (principal value) ex.

14. Complex roots and powers (복소수 근과 멱수) - For complex a and b, - Since ln a is multiple values, powers a^b are usually multiple values (cf. principal value). ex. 1 Find all values of i^(-2i)

15. Inverse trigonometric and hyperbolic functions (역삼각함수와 역쌍곡함수)

16. Some applications (응용) - Electricity (method 1) ‘complicated function’

(method 2)