{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

complex080521

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

This preview shows pages 1–3. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 8

complex080521 - Phys374 Spring 2008 Prof Ted Jacobson...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online