Math132-ComplexNumbersandFunctions数学132复杂的数字和功能.doc

Math 132 - Complex Numbers and FunctionsMany engineering problems can be treated and solved by using complex numbers and complex functions. We will look at complex numbers, complex functions and complex differentiation. Part I Complex Numbers In algebra we discovered that many equations are not satisfied by any real numbers. Examples are:or We must introduce the concept of complex numbers. Definition: A complex number is an ordered pair of real numbers x and y. We call x the real part of z and y the imaginary part, and we write , .Example 1: and Two complex numbers are equal where and : if and only if and Addition and Subtraction of Complex Numbers: We define for two complex numbers, the sum and difference ofand : and .Multiplication of two complex numbers is defined as follows:Example 2: Let and then and and . We need to represent complex numbers in a manner that will make addition and multiplication easier to do. Complex numbers represented as A complex number whose imaginary part is 0 is of the form and we have and and which looks like real addition, subtraction and multiplication. So we identify with the real number and therefore we can consider the real numbers as a subset of the complex numbers. We let the letter and we call i a purely imaginary number. Now consider and so we can consider the complex number = the real number . We also get And so we have:Now we can write addition and multiplication as follows: Example 3: Let and , then and and = .The Complex PlaneThe geometric representation of complex numbers is to represent the complex number as the point . y-axis 2 1 1 2 x-axis So the real number is the point on the horizontal x-axis, the purely imaginary number is on the vertical y-axis. For the complex number , is the real part and y is the imaginary part.Example 4. Locate 2-3i on the graph above. How do we divide complex numbers? Lets introduce the conjugate of a complex number then go to division. Given the complex number , define the conjugate We can divide by using the following:Example 5. Problem Set I Find 1. 2. 3. 4. 5. 6. 7. 8. 9. Let and and , find10. 11. 12. 13. 14. 15. Graph the following: 16. 17. 18. and its conjugate.19. Find the solutions of.20. Find the solution of.Complex Numbers in Polar Form It is possible to express complex numbers in polar form. If the point is represented by polar coordinates , then we can write , and . r is the modulus or absolute value of z, , and q is the argument of z, . The values of r and q determine z uniquely, but the converse is not true. The modulus r is determined uniquely by z, but q is only determined up to a multiple of 2p. There are infinitely many values of q which satisfy the equations , but any two of them differ by some multiple of 2p. Each of these angles q is called an argument of z, but, by convention, one of them is called the principal argument. DefinitionIf z is a non-zero complex number, then the unique real number q, which satisfies is called the principal argument of z, denoted by .Note:The distance from the origin to the point is , the modulus of z; the argument of z is the angle . Geometrically, q is the directed angle measured from the positive x-axis to the line segment from the origin to the point . When , the angle q is undefined.The polar form of a complex number allows one to multiply and divide complex number more easily than in the Cartesian form. For instance, if then , . These formulae follow directly from DeMoivres formula.yxExample 6. For , we get and . The principal value of is , but would work also. Multiplication and Division in Polar FormLet and then we haveand Example 7: and Then =Since And= We can use =And so: DeMoivres Theorem: where n is an positive integer.Let to get: .Example 8: Compute Roots of Complex Numbers:Consider = (Equation 1)where . Then , and so or . However also satisfies Equation 1 and so . And implies . However implies . And continuing implies . for k any integer up to n. We get ,k=0, 1, 2, 3, , (n-1).Example 9:Find the square roots of i.Since , we let is one square root of . The second square root of is :.Example 10. Find the sixth root of There will be six roots:Example 11: Compute Solution: =Where . So =.Problem Set IIWrite in polar form:1. 2. 3. Write in rectangular form:4. 5. 6. 7. Find 8. Find 9. Find 10. Find Part II Functions, Neighborhoods and Limits We consider the concept of a function of a complex number. For , where x and y are real numbers, we know about limits, continuity and derivatives. Let the complex number , where and . Here x and y are independent variables and where and . This is an example of a complex valued function, w, of a complex variable z. In general, letwhere and z Distance in the complex plane. In the complex plane let and then is the distance between the complex numbers and Note that is the distance from the origin to z. Definition: A neighborhood of the point in the complex plane is the set of points where and Definition: A function is said to have a limit L as approaches ( and written ) iff is defined in a neighborhood , except at and the values of f are close to L for all z close to . Mathematically: for every , we can find such that for all in the neighborhood , so that if , then . Notice that this definition of a limit is similar to the definition in calculus. The difference here is that z can approach from any direction in the complex plane. Definition: A function is continuous at if is defined and . By definition a complex function that is continuous at is defined in a neighborhood of . is continuous in a domain D if it is continuous at each point in the domain D. The derivative of a function at a point z is defined as: provided that limit exists. Example 12: Let . To compute we consider The rules for derivatives are the same for calculus: c=constant Definition: A function is defined to be analytic in the domain D if is defined and has a derivative at all points of D. The Exponential Function The definition of the exponential function is given in terms of the real functions , , and To show why this is true, consider the power series expansion: =and substitute for x, to get: =+=. The formula is known as Eulers formula. Theorem: is never zero.Proof:, therefore, cannot be zero.Theorem:If y is real, then .Proof: and , which implies .Theorem: if and only if , where n is an integer.Proof:If , . Conversely, suppose , then . Since , this implies ; andwhere k is an integer. Substituting this into implies . Hence k is even, since . Therefore, .Theorem: if and only if , where n is an integer.Proof: if and only if . Then use the previous theorem.Definition of and To get the definitions of and we substitute for y in Eulers Formula to get:=+=.Adding and together and dividing by 2, we get: .Subtracting from , and dividing by , we get: .We now define and.Substitute ix for y in to get .Similarly we get .Facts: 1. For , to get 2. The derivative of is . 3. 4. For we get . 5. In polar form . 6. It now follows that . 7. 8. 9. Logarithmic FunctionsIf then we write , called the natural logarithm