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MATH23 MULTIVARIABLE CALCULUS

COMPLEX NUMBERS. MATH23 MULTIVARIABLE CALCULUS. OBJECTIVES At the end of this lesson, the students are expected to : Define Complex Numbers Differentiate Imaginary from Real Numbers Perform Algebraic Manipulations with Complex Numbers Identify Properties of Complex Numbers

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MATH23 MULTIVARIABLE CALCULUS

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  1. COMPLEX NUMBERS MATH23 MULTIVARIABLE CALCULUS

  2. OBJECTIVES At the end of this lesson, the students are expected to : Define Complex Numbers Differentiate Imaginary from Real Numbers Perform Algebraic Manipulations with Complex Numbers Identify Properties of Complex Numbers Perform Trigonometric, Logarithmic and Exponential Functions of Complex Numbers

  3. COMPLEX NUMBERS The first to use complex numbers for solutions to equations was the Italian mathematician Girolamo Cardano, who found the formula for solving cubic equations. The term “complex number” was introduced by the great German mathematician Carl Friedrich Gauss, who also paved the way for a general and systematic use of complex numbers.

  4. COMPLEX NUMBERS DEFINITION: A number of the form of , a + b i with a and b are real constants and is called a complex number. The number a is called the real part, and b is called the imaginary part .

  5. EQUALITY OF COMPLEX NUMBERS PROPERTY: Two complex numbers are equalif and only if the real parts are equal and the imaginary parts are also equal. That is, a + bi = c + di if and only if a = c and b = d Example: If 3x – 5 + ( 4x + 3)i = y – 4 + ( 3x + y)i, find x & y. Solution: Real: 3x-5 = y -4 Imaginary: 4x+3 = 3x+y Hence solving for x and y: x= 2 and y = 5

  6. EXAMPLES Solve the following equations for x and y. 1. 4 – xi = y – 3i  Answer: x = 3 and y =2 2. 2y – 3 + ( 4x + 8)i = 0  Answer: x = -2 and y =3/2  3. 3x + ( y – x)i = 6  Answer: x = 2 and y =2  4. x + 2y + 3i = 3 + ( 2x – y)i   Answer: x = 9/5 and y =3/5  5. 2x + i log y = 8 + 3i Answer: x = 3 and y =1000

  7. POWERS OF (i) From In general, i n = 1 if n is divisible by 4 in = -1 if n is divisible by 2 but not divisible by 4 = i if n is divided by 4, the remainder is 1 = -i if n is divided by 4 the remainder is 3

  8. EXAMPLES Answers: (a). i (b). -1 (c). -i Simplify: (a). i234 - i2789 (b). i2563 + i346465 (c). 3i22415+2 i2536 Simplify: (a). (b). (c). Answers: (a). -1-i (b). 0 (c). 2-3i

  9. OPERATIONS ON COMPLEX NUMBERS SUBTRACTION Let z = a +biand w = c + di be elements of the set of complex number. Then the following operations are: ADDITION MULTIPLICATION

  10. OPERATIONS ON COMPLEX NUMBERS DIVISION Let z = a +biand w = c + di be elements of the set of complex number. Then the following operations : CONJUGATE

  11. OPERATIONS ON COMPLEX NUMBERS Perform the indicated operations, leaving each result in the form a + bi. ( 5 – i) – ( 3 + 2i) + ( 2 + i) 2. i + ( 8 – 3i) + (- 2 +7 i) Answer: 4- 2i Answer: 6+5i 3. ( 3 + 2i)( 2 -3 i) 4. ( 4 – 5i)( 4 + 5i) Answer: 12-5i Answer: 41 5. ( 4 – 3i)2 6. ( 4 - 3i)3 Answer: 7-24i Answer: -44-117i 7. 8. Answer: (-4+3i)/25 Answer: (7-4i)/13

  12. GRAPHICAL REPRESENTATION Consider the rectangular form of the complex number z = a + bi then z = r cos  + r sin  i z = r ( cos + i sin  ) = r cis, this is called the polar form of z. When a rectangular coordinate plane is used to represent complex numbers, the x-axis is called the real axis, the y-axis the pure imaginary axis and the plane the complex plane. Fig 1 Argand Diagram

  13. RECTANGULAR TO POLAR FORM The Rectangular Form of the Complex Number is of the form: z = x + y i Since : x = r cos q y = r sin q This can be expressed equivalently as z = (r cos q) + i (r sin q) z = r ( cos q + i sin q) z = r cis q = r  where r = is called the modulus and  = Tan-1 (b/a) is called the argument of z

  14. POLAR TO RECTANGULAR FORM The Polar Form of the Complex Number is of the form: z = r cis q = r  Since :  = Tan-1 (b/a) and z =

  15. EXPONENTIAL FORM The Exponential Form of the Complex Number is of the form: z = r e q i Proof: The Euler’s Formula eui= cos u + i sin u

  16. PRODUCT or QUOTIENT IN EXPONENTIAL FORM When the complex number is expressed in polar form: The product can be expressed as: And the quotient as: When the complex number is expressed in exponential form: The product and the quotient can be expressed as: can be expressed as:

  17. EXAMPLES Express the results in a + bi can be expressed as:

  18. DE MOIVRE’S THEOREM If n is any positive integer, then the nthpower of the complex number z = a + bi = r cis  is given by z n= r n cis n or equivalently as, ressed as:

  19. EXAMPLES [2(cos 35o + isin 35o)]3 Express the results in a + bi can be expressed as: • (1 - i)4

  20. ROOTS OF COMPLEX NUMBERS If n is any positive integer, then the nthroot of the complex number z = a + bi = r cis  is given by Where k is 0, 1 , 2…n ressed as:

  21. EXAMPLES Express the results in a + bi The square roots of 16 cis 80o. The cube roots of 27 cis 150o The fourth roots of 81 cis 160o Find the roots of x16+81 =0 x6 +4x3+3= 0

  22. TRIGONOMETRIC FUNCTIONS OF COMPLEX NUMBERS The trigonometric function of imaginary number is defined as: And recalling the trigonometric identity: Therefore:

  23. HYPERBOLIC FUNCTIONS OF COMPLEX NUMBERS From the trigonometric function of imaginary number: Then: sinh( bi) = i sin b cosh( bi) = cos b And recalling the trigonometric identity: Therefore:

  24. EXPONENTIAL FUNCTIONS OF COMPLEX NUMBERS If z = a + bi then the exponential function of z is e z = e a + bi simplifying ez = e ae bi Examples: e2+3i 2. e4 cis 2 3. 3e2 cis 3 4. 42+i

  25. NATURAL LOGARITHM OF COMPLEX NUMBERS From the general exponential form z= re (  + 2k )i where  is in radians and taking the natural logarithms of both members we get ln z = ln r + (  + 2k )i When k = 0 then the principal value of ln z= ln r + i

  26. EXAMPLES Evaluate the following: 1. 2. Log i (2+7i)

  27. COMPLEX RAISED TO COMPLEX NUMBERS If z = a + bi and w = c + di then z w = (a + bi)c + di Supposez = re i = e ln r e i = e ( ln r +  i) Then zw = (eln r +  i)(c + di)  Let p + qj= (ln r + i)(c + di) Thus zw = e p + qi = e pe qi = e pcis q

  28. EXAMPLES Evaluate the following: 1. (2+3i) (1+i) 2. (2-i) i 3. ii 3. If x i = i , Find x.

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