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CalcIVComplex1 - Informal lecture notes for Nov 27 I'll...

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Informal lecture notes for Nov. 27 I’ll assume you’re familiar with the review of complex numbers and their algebra as contained in Appendix G of Stewart’s book, so we’ll pick up where that leaves off. 1 Elementary complex functions In one-variable real calculus, we have a collection of basic functions, like poly- nomials, rational functions, the exponential and log functions, and the trig functions, which we understand well and which serve as the building blocks for more general functions. The same is true in one complex variable; in fact, the real functions we just listed can be extended to complex functions. 1.1 Polynomials and rational functions We start with polynomials and rational functions. We know how to multiply and add complex numbers, and thus we understand polynomial functions. To be specific, a degree n polynomial, for some non-negative integer n , is a function of the form f ( z ) = c n z n + c n - 1 z n - 1 + · · · + c 1 z + c 0 , where the c i are complex numbers with c n = 0. For example, f ( z ) = 2 z 3 + (1 - i ) z +2 i is a degree three (complex) polynomial. Polynomials are clearly defined on all of C . A rational function is the quotient of two polynomials, and it is defined everywhere where the denominator is non-zero. Example: The function f ( z ) = z 2 +1 z 2 - 1 is a rational function. The denomina- tor will be zero precisely when z 2 = 1. We know that every non-zero complex number has n distinct n th roots, and thus there will be two points at which the denominator is zero. It’s easy to see that those points are 1 and - 1, and so f is defined on C \ { 1 , - 1 } . If we want to compute f at some point, we just use our rules for complex algebra. For instance, using that (1 + i ) 2 = 1 + 2 i - 1 = 2 i , we have f (1 + i ) = (1 + i ) 2 + 1 (1 + i ) 2 - 1 = 1 + 2 i - 1 + 2 i = 1 + 2 i - 1 + 2 i · - 1 - 2 i - 1 - 2 i = - 1 - 2 i - 2 i + 4 1 + 4 = 3 5 - 4 5 i. 1.2 The exponential function and logarithm We’ve already seen the (complex) exponential function. Recall that we have the useful formula e x + iy = e x (cos y + i sin y ) . 1
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Here we use our usual convention of writing a complex number as z = x + iy for real x and y . One reason this formula is useful is that it allows us to actually compute. We had originally defined e z by a power series, and if we used that definition directly, then evaluating something like e 1+ πi would require summing an infinite series. Instead, using the above formula allows us to write e 1+ πi = e 1 (cos π + i sin π ) = - e. Another reason this formula is useful is that it allows us to give a somewhat intuitive interpretation of the exponential function. Notice that we have written our “input” variable in Cartesian coordinates, but our “output” variable is in polar coordinates. We see that e x + iy is the complex number with modulus (or r , in polar coordinates) e x and argument (or θ , in polar coordinates) y . If our input is a real number, then y is zero and the output will be real and positive (since the positive real axis has argument zero) and will have modulus e x . Thus we confirm that this agrees with the real exponential function for real input.
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