COMPLEX EIGENVALUES Math 21b, O. Knill HOMEWORK: Section 7.5: 8,12,24,32,38,60*,36* NOTATION. Complex numbers are written as z = x + iy = r exp( iφ ) = r cos( φ ) + ir sin( φ ). The real number r = | z | is called the absolute value of z , the value φ is the argument and denoted by arg( z ). Complex numbers contain the real numbers z = x + i 0 as a subset. One writes Re( z ) = x and Im( z ) = y if z = x + iy . ARITHMETIC. Complex numbers are added like vectors: x + iy + u + iv = ( x + u ) + i ( y + v ) and multiplied as z * w = ( x + iy )( u + iv ) = xu-yv + i ( yu-xv ). If z 6 = 0, one can divide 1 /z = 1 / ( x + iy ) = ( x-iy ) / ( x 2 + y 2 ). ABSOLUTE VALUE AND ARGUMENT. The absolute value | z | = p x 2 + y 2 satisFes | zw | = | z | | w | . The argument satisFes arg( zw ) = arg( z ) + arg( w ). These are direct consequences of the polar representation z = r exp( iφ ) , w = s exp( iψ ) , zw = rs exp( i ( φ + ψ )). GEOMETRIC INTERPRETATION. If z = x + iy is written as a vector ± x y ² , then multiplication with an other complex number w is a dilation-rotation : a scaling by | w | and a rotation by arg( w ). THE DE MOIVRE ±ORMULA. z n = exp( inφ ) = cos( nφ ) + i sin( nφ ) = (cos( φ ) + i sin( φ )) n follows directly from z = exp( iφ ) but it is magic: it leads for example to formulas like cos(3 φ ) = cos( φ ) 3-3 cos( φ ) sin 2 ( φ ) which would be more di²cult to come by using geometrical or power series arguments. This formula is useful for example in integration problems like R cos( x ) 3 dx , which can be solved by using the above deMoivre formula. THE UNIT CIRCLE. Complex numbers of length 1 have the form
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