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Extra info for An Introduction to Complex Analysis for Engineers
The simplest answer is `well it jolly well ought to be', but if you take z = 1 and square it and then take the square root, there is no particular reason to insist on taking the positive value. On the other hand, suppose we adopt the convention that we mean the positive square root for positive real numbers, in other words, on the positive reals, square root means what it used to mean. Are we forced to take the negative square root for negative numbers? No, we can take any one we please. But suppose I apply two rules: 1.
And if you can't, then you just have to plug away doing algebra, with a serious risk of making a slip and wasting hours of your time as well as getting the wrong answer. Seeing the patterns that make things happen the way they do is quite interesting, and it is boring to just plug away at algebra. So it is worth a bit of trouble trying to understand the stu as opposed to just memorising rules for doing the sums. If you can cheerfully hop to the matrix representation of complex numbers, some things are blindingly obvious that are completely obscure if you just learn the rules for multiplying complex numbers in the classical form.
Who. The origin does have something special about it, but it is the only point that does. The attempt to choose regions which are restricted in angular extent so that you can get a one-one map for the squaring function and so choose a particular square root is harmless, but it seems odd to call the resulting bits `branches'. ) It is entirely up to you how you choose to do this cutting up of the space into bits. Of course, once you have taken a region, squared it, conrmed that the squaring map is one-one and taken your inverse, you still have to reckon with the fact that someone else could have taken a dierent region, squared that, and got the same set as you did.
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