complex080521 - Phys374, Spring 2008, Prof. Ted Jacobson...

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Phys374, Spring 2008, Prof. Ted Jacobson Department of Physics, University of Maryland Complex numbers—version 5/21/08 Here are brief notes about topics covered in class on complex numbers, focusing on what is not covered in the textbook. 1 Complex algebra and geometry 1. i as a solution to x 2 + 1 = 0, that is, i = - 1. 2. Complex numbers: z = x + iy , with x and y real numbers, called the real and imaginary parts of z , x = Re ( z ) and y = Im ( z ). 3. Square roots of other negative numbers. E.g. - 2 = i 2. 4. Any quadratic equation az 2 + bz + c = 0 has two roots, given by z = ( - b ± b 2 - 4 ac ) / 2 a . If b 2 - 4 ac is negative, the roots are complex. 5. Fundamental theorem of algebra : Every polynomial has at least one root. This easily implies by induction that every n th order polynomial can be factorized as a n ( z - w 1 )( z - w 2 ) ··· ( z - w n ). That is, it has n roots, some or all of which may coincide. For a proof of this see below. 6. Complex conjugate : z * = x - iy . 7. Inverse of a complex number: 1 z = z * z * z = x - iy x 2 + y 2 . (1) 8. Modulus : | z | = z * z = p x 2 + y 2 . 9. Everything is nice: ( z + w ) * = z * + w * , ( zw ) * = z * w * , (1 /z ) * = 1 /z * , | zw | = | z || w | , | z * | = | z | . 10. Exponential function : The exponential function f ( x ) = e x can be defined as the unique function equal to its own derivative and equal to 1 when x = 0. To find its Taylor series, write f ( x ) = 1 + a 1 x + a 2 x 2 + a 3 x 3 + ··· and impose equality of df/dx and f for all x . The coefficients of x n must then be equal, which implies a n = 1 /n !. Thus e x = n =0 x n /n !. This definition can be generalized to any complex number z , replacing x by z . 1
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11. Key identity: e a e b = e a + b . One can prove this by writing out each of the series expansions, multiplying term by term on the left hand side, and expanding ( a + b ) n on the right hand side, then collecting terms with equal powers of a and b on both sides. A much simpler proof: define f ( a ) = e a e b and g ( a ) = e a + b , and note that df/da = f and dg/da = g . Also f (0) = g (0). So f and g are equal at one point and satisfy the same first order differential equation, so they are equal everywhere! 12. Complex plane : Complex numbers can be viewed as points or vec- tors on a plane, with Re ( z ) on the horizontal axis and Im ( z ) on the vertical axis. The distance from the origin to z is | z | . 13. Euler’s identity: e = cos θ + i sin θ . Two different proofs: (a) Expand both sides in a power series and show they are equal term by term, or (b) note that both sides are equal to i times their own derivative with respect to θ , so they satisfy the same first order differential equa- tion. They are also obviously equal when θ = 0, so they are equal everywhere. 14.
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This note was uploaded on 12/29/2011 for the course PHYSICS 374 taught by Professor Jacobson during the Fall '10 term at Maryland.

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complex080521 - Phys374, Spring 2008, Prof. Ted Jacobson...

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