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Appendix1

Appendix1 - 1 APPENDIX 1 COMPLEX NUMBERS AND PROBABILITY A...

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1 APPENDIX 1: COMPLEX NUMBERS AND PROBABILITY A. Complex numbers and wave interference The number i satisfies i 2 = 1. Let us write a general complex number in cartesian and polar forms as z = x + iy = re , where x = Re ( z ) and y = Im ( z ) are real numbers, called the real and imaginary parts of z , respectively; r 0 is the modulus and θ the argument (or phase) of z . The complex conjugate of any complex expression is found by replacing i → − i in the expression, e.g., z = x iy is the complex conjugate of z . Since by simple geometry in the Argand plane x = rcosθ and y = rsinθ, we have e = cosθ + isinθ . By using this formula and its complex conjugate we find the important results, e + e = 2 cosθ and e e = 2 isinθ . Modulus : Recall | z | 2 = zz 0 . In terms of the cartesian representation, | z | = | x + iy | = radicalbig x 2 + y 2 0 . The modulus of a product is the product of the moduli, | z 1 z 2 | = | z 1 || z 2 | (check: | re | = | r || e | = r × 1 , as it should.) The modulus of a sum of complex numbers is important in understanding interference of waves, i.e., the principle of superposition for linear waves: | z 1 + z 2 | 2 = ( z 1 + z 2 )( z 1 + z

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Appendix1 - 1 APPENDIX 1 COMPLEX NUMBERS AND PROBABILITY A...

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