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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 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 i , where x = Re ( z ) and y = Im ( z ) are real numbers, called the real and imaginary parts of z , respectively; r is the modulus and 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 = rcos and y = rsin, we have e i = cos + isin . By using this formula and its complex conjugate we find the important results, e i + e i = 2 cos and e i e i = 2 isin . Modulus : Recall | z | 2 = zz . In terms of the cartesian representation, | z | = | x + iy | = radicalbig x 2 + y 2 . The modulus of a product is the product of the moduli, | z 1 z 2 | = | z 1 || z 2 | (check: | re i | = | r || e i | = 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 superpositionof a sum of complex numbers is important in understanding interference of waves, i....
View Full Document

Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online