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18.03 LECTURE NOTES, SPRING 2014 - MIT Mathematics

LECTURE NOTES, SPRING 2014. BJORN POONEN. 7. complex numbers complex numbers are expressions of the form x + yi, where x and y are real numbers, and i is a new symbol. Multiplication of complex numbers will eventually be defined so that i2 = 1. (Electrical engineers sometimes write j instead of i, because they want to reserve i for current, but everybody else thinks that's weird.) Just as the set of all real numbers is denoted R, the set of all complex numbers is denoted C. Flashcard question: Is 9 a real number or a complex number ? Possible answers: 1. real number 2. complex number 3. both 4. neither Answer: Both, because 9 can be identified with 9 + 0i. Operations on complex numbers. real part Re(x + yi) := x imaginary part Im(x + yi) := y (Note: It is y, not yi, so Im(x + yi) is real).

one could not use it to de ne ez, because \eraised to a complex number" has no a priori meaning. Theorem 7.5. The complex exponential function ez has the following properties: (a) The derivative of e zis e. (b) e0 = 1. (c) ez+ w= eze for all complex numbers zand w. (d) (ez)n = enz for every complex number zand integer n. The n= 1 case says

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