Rough Complex Crash Course !

Kept Girl - Complex Analysis Crash Course 1 Dorothy Moorefield April 22, 2009 What is i ? Take a real number and let’s call it, x. Definition 0.1 Then a number, b, is the square root of x, if b2 = bb = x. For example, 2 is the square root of 4 because 22 = 4. 9 = 32 and thus 3 is the square root of 9. Consider a negative number, like −1. What number times itself gives us −1? 1 times itself is 1 so that doesn’t work so it must be −1 right? Now I bet all of you folks that remember your pre-algebra are fussing at me right now saying... but, but, a negative times a negative gives a positive. Yeah, that’s an issue. So (−1)2 = 1. Is there any real number that is the square root of −1? Turns out the answer to that is no. The square root of a negative number is not defined in the real number system. However, we are not going to let this problem stop us! Clearly the best way to deal with this problem is to create a whole new type of number. This is where i comes in. By definition, i is the square root of −1. Now that we have found this new type of number, it makes since to ask if there are any other numbers out there like i , or is i all by itself? Let’s consider −4. What is the square root of −4. As with −1, the square root is not 2 or −2. Consider 2i . We have (2i )2 = (2i )(2i ) = 22 i 2 = 4i 2 . Now that we know i 2 = −1, we can conclude 4i 2 = 4(−1) = −4. Thus, 2i is the square root of −4. It turns out that with our new friend i , we can now find a square root for every real number. Let’s give our new numbers a home. Complex Numbers Definition 0.2 Let a and b be two real numbers, and i be the square root of −1. A complex number is a number of the form a + bi . The real number, a is referred to as the real part of the complex number and the real number b is referred to as the imaginary part of the complex number. Now that we have our new numbers, let’s check out their arithmetic properties. Let α = a + bi and γ = c + di . Then α + γ = (a + bi ) + (c + di ) = (a + c) + bi + di = (a + c) + (b + d)i . One can see that the addition of two complex numbers can be contained by adding the real parts together and adding the imaginary parts together. Recall from basic algebra how to multiply two polynomials. Now lets check out the multiplication of complex numbers. αγ = (a + bi )(c + di ) = ac + adi + (bi )c + (bi )(di ) = ac + (ad + bc)i + bdi 2 = ac − bd + (ad + bc)i. Now notice that real numbers are actually complex numbers. Let a be a real number. Then a = a+0i . Kinda cool. Also, complex numbers satisfy the same properties of addition and multiplication that 1

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hold in the real number system. I’ll leave that as an exercise for folks to show. If you want to post it up or have any questions feel free to post a comment on the blog. Useful definitions and a couple of theorems Let α = a + bi be a complex number. Definition 0.3 The conjugate of α, denoted by, α ¯ , is defined to be α ¯ := a − bi . Note αα ¯ = a2 + b2 . Lots of my students just memorized that the absolute value of a real number is just the positive value of the number without really understanding why. It turns out that the absolute value of a real number is it’s distance from 0 on the real number line. I will refer you to a link on the complex plane so you can see pictures, which are often very helpful. Meanwhile, Definition 0.4 The absolute value of a complex number, α = a + bi is it’s distance from the origin of the complex plane given by p |α| = a2 + b2 Theorem 0.5 Let α and γ be two complex numbers then |αγ| = |α||γ| and |α + γ| ≤ |α| + |γ| The later part is what is know as the triangle inequality, which is a very important and useful property in both complex and real analysis. This concludes my synopsis of Section 1.1 of Serge Lang’s ”Complex Analysis”. The exercises in this section are pretty straight forward arithmetic manipulations. I do recommend picking up the book and working through them, however, I most likely won’t be posting up any solutions from this section. More later...