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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).

existence and uniqueness theorem shows that there is a unique such function f(z) satisfying f0(z) = f(z); f(0) = 1: This function is called the complex exponential function ez. The number eis de ned as the value of ez at z= 1. But it is the function ez, not the number e, that is truly important. De ning ewithout de ning ez rst is a little ...

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