The Chemistry Maths Book, Second Edition

230 Chapter 8Complex numbers EXAMPLES 8.5Expressz 1 = 1 x 1 + 1 iyin polar form. (i) z 1 = 111 + 1 i We havex 1 = 11 andy 1 = 11 ...

8.3 Graphical representation 231 is obtained by completing the parallelogram OPSQ. Point S has coordinates (x 1 1 + 1 x 2 , y 1 ...

232 Chapter 8Complex numbers If the two numbers are a complex conjugate pair,z 1 = 1 x 1 + 1 iyandz* 1 = 1 x 1 − 1 iy, then arg ...

8.3 Graphical representation 233 (ii) θ 1 1 − 1 θ 2 1 = 1 − 13 π 212 and, by equation (8.22), (iii)z 2 2 z 1 1 = 1 (z 1 2 z 2 ) ...

234 Chapter 8Complex numbers This is de Moivre’s formulafor positive integers n. 3 The formula is also valid for other values of ...

8.4 Complex functions 235 Therefore, cos 15 θ 1 = 1 cos 5 1 θ 1 − 1101 cos 3 1 θ 1 sin 2 1 θ 1 + 151 cos 1 θ 1 sin 4 1 θ 1 = 116 ...

236 Chapter 8Complex numbers A more advanced application of complex numbers is in the extension of the concepts of variable and ...

8.5 Euler’s formula 237 This relation between the exponential function and the trigonometric functions is called Euler’s formula ...

238 Chapter 8Complex numbers The number can therefore be written as with complex conjugate and inverse Also, 0 Exercises 30–33 E ...

8.5 Euler’s formula 239 de Moivre’s formula When θin Euler’s formula, (8.33), is replaced by nθwhere nis an arbitrary number, th ...

240 Chapter 8Complex numbers The coordinates of the rotated point are therefore related to the coordinates of the unrotated poin ...

8.6 Periodicity 241 so that each is a cube root of the number 1: z 0 1 = 1 e 0 1 = 11 We note thatz 1 andz 2 are a complex conju ...

242 Chapter 8Complex numbers A function that has the same circular periodicity as the figure with nequidistant points on a circl ...

8.6 Periodicity 243 system is called a ‘rigid rotor’ and the equation of motion in quantum mechanics (the Schrödinger equation) ...

244 Chapter 8Complex numbers We note that the quantization of the energy of the system has arisen as a consequence of applying t ...

8.5 Exercises 245 The integralIis the real part of this: (8.55) The imaginary part is a bonus: (8.56) 0 Exercises 49, 50 8.8 Exe ...

246 Chapter 8Complex numbers Section 8.4 (i)Express the complex functionf(x) 1 = 13 x 2 1 + 1 (1 1 + 12 i)x 1 + 1 2(i 1 − 1 1) ...

9 Functions of several variables 9.1 Concepts When the equation of state of the ideal gas is written in the form it is implied t ...

248 Chapter 9Functions of several variables 9.2 Graphical representation We saw in Section 2.2 that a (real) function of one var ...

9.3 Partial differentiation 249 Each of these graphs is a planar ‘cut’ through the three-dimensional surface. In the general cas ...