Branch points and branch cuts

Branch points and branch cuts - Branch points and branch...

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Branch points and branch cuts From Physics Notes When we introduced complex algebra, we postponed discussion of what it means to raise a complex number to a non-integer power, such as , , or . It is now time to open that particular can of worms. Out of this can will come two indispensible concepts: branch points and branch cuts , which will be very important for studying functions that map complex inputs to complex outputs. Contents 1 Taking non-integer powers 1.1 Complex logarithms 2 Dealing with multi-valued operations 2.1 Branch cuts 2.2 Branch points 3 Branch cuts for general multi-valued operations 3.1 Example 4 Exercises Taking non-integer powers Given a complex number in its polar representation, , raising to the power of could be handled this way: The problem lies in the complex exponential term . The variable is an angle, so if we change it by any integer multiple of , the complex number is unchanged. If this fact is taken into account, the above equation becomes If the power is an integer, there's no problem: will be an integer multiple of , and hence the operation gives a unique answer. On the other hand, if is not an integer, then there is no unique answer. In this case, the "raise to the power of " operation is a multi-valued operation . It cannot be treated as a function in the usual sense, since a function needs to produce unique well-defined outputs. Let's take a closer look at the multiple values. If is irrational, never repeats itself modulo . This means we have an infinite number of different results for (one for each integer ). In principle, each of these results is equally legitimate. More interesting is the case of a non-integer rational power, which can be written as , where and are integers with no common divisor. Then has unique values modulo :
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Fig. 1: Roots of unity for . (Note that this set is independent of ; you can check for yourself that only affects the sequence in which these numbers are generated, which is unimportant for our present discussion.) Hence, where can take any of the values listed above. The values of are called roots of unity , and they are located in the complex plane at evenly-spaced positions along the unit circle, as shown in Fig. 1. Example Consider the complex "square root" operation, . If we write in its polar respresentation, then The factor of takes on two possible values: for odd , and for even . Hence, the square root operation has two possible values:
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Example Consider the operation . Writing in its polar respresentation: The ambiguity now arises in the factor of . We can work out the value of this factor for different values of the integer : Hence, Complex logarithms Here is another way to think about the problem of non-integer powers. Recall what it means to raise a number to, say, the power of 5: we simply multiply the number by itself five times. How can we define raising a number to a non-integer power ? When discussing real numbers, the definition we took was based on a combination of exponential and logarithm functions: This definition relies on the fact that, for real inputs, the logarithm is a valid (well-defined) function. That, in turn,
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