Questions 14 - 29

These questions examine the mathematical properties of complex numbers.

Q14

Calculate \(e^i\).

Q15

Find the real and imaginary parts of

(a) \(ie^{-ix}\), (b) \(e^{in\pi}\), and (c) \(e^{in\pi/2}\),

where \(n\) is an integer.

Q16

Calculate (a) \(i^i\) and (b) \(i^{1/i}\).

Strategy: Using different bases such as \(a^x = e^{x \ln(a)}\) any number can be raised to any power. With complex numbers always try to put the number in terms of Euler’s equation.

Q16

The cosine function is defined as \(\displaystyle \cos(x) = \frac{e^{ix}+e^{-ix}}{2}\). What is \(\cos^{-1}(x)\) ?

Strategy: This is a case where \(x\) and \(y\) are swapped about. If \(\cos^{-1}(x)\) then \(\cos(y) = x\). It is true also that \(\displaystyle \cos(y) = \frac{e^{iy}+e^{-iy}}{2}\). Next eliminate the cosine and solve for \(y\) and so find the answer.

Q18

Show that the identity \((\cos(x) + \sin(x))^2 = 1 + sin(2x)\) can be (relatively easily) proved using complex numbers.

Q19

Show that \(\displaystyle 2\sin\left( \frac{a+b}{2} \right)\cos\left( \frac{a-b}{2} \right) =\sin(a)+\sin(b)\)

Q20

Starting with Euler’s theorem and letting \(\theta = a + b\), calculate \(\sin(a + b)\) and \(\cos(a + b)\) by equating real and imaginary parts.

Q21

Find \(\sin(\theta)\) in exponential form then calculate \(| \sin(i\theta) |\), and compare it with \(| \sin(\theta) |\). Plot values of \(|\sin(ix) |\) and \(|\sin(x) |\) over the range \(x = -4 \cdots 4\).

Q22

(a) If \(z = \cos(x) + i \sin(x)\) show that \(dz/dx = iz\).

(b) Integrate this result and prove Euler’s theorem.

Q23 Fourier transform of a square wave

Calculate the real and imaginary parts of \(\displaystyle \frac{1}{\sqrt{2\pi}}\left( \frac{1-e^{i\omega t}}{i\omega} \right)\).

This function is the Fourier transform of a square wave of length \(t\). See chapter 9.5, and 9.6.

Strategy: use \(i = -1/i\) and multiply out the terms.

Q24 FID in NMR

In an NMR experiment, the FID signal has the form \(\displaystyle s(t) = \sum_j a_je^{i\omega_j t-t/\tau_j}\) where \(\omega\) is the frequency of the transition, \(\tau\) the average of the T\(_1\) and T\(_2\) lifetimes, and \(a\) the amplitude of each signal and there are \(j\) parts to the total signal. For simplicity, assume that \(\tau_j\) has a constant value \(\tau\).

(a) Calculate the real, imaginary, and absolute value of \(s\) if \(j = 2\).

(b) Plot the real part of the signal if \(a_1 =a_2 =2\) and \(2\pi\omega =1\) Hz and also when \(0.2\) Hz and \(\tau_1 =\tau_2 =50\) s and also when \(\tau_1 = \tau 2 = 500\) s. Comment on the two results.

(c) Repeat (a) when \(a_1 = i\), which means that the initial amplitude is complex, and \(a_2 = 1\).

In spite of the fact that the signal from an experiment cannot be a complex number, this is what appears to be the case here. The reason for this is that in a real NMR experiment two signals are measured, one by a coil on the spectrometer’s x-axis and the other by a similar coil on the y-axis. These are at right angles to the z-axis along which the permanent magnetic field is directed. These x and y signals are measured in quadrature, i.e. 90\(^\mathrm{o}\) out of phase to one another. One signal is taken to be the real component, and one the imaginary. They are then combined to produce \(s(t)\) given above.

Strategy: The question asks you to find the components which when combined make \(s(t)\). Use the Euler formula to do this and to simplify the complex exponential. As the signal represents the FID from an NMR experiment it oscillating in a sinusoidal way.

Q25

Derive the identities

(a) \(4 \cos(\theta)\sin^2(\theta) = \cos(\theta) - \cos(3\theta)\)

(b) \(4 \sin(\theta)\cos^2(\theta) = \sin(\theta) + \sin(3\theta)\)

Strategy: Always use the exponential forms of sine and cosine wherever possible for complicated trig functions. These are \(\displaystyle \cos(\theta)=\frac{e^{i\theta}+e^{-i\theta}}{2}, \qquad \sin(\theta)=\frac{e^{i\theta}-e^{-i\theta}}{2i}\)

Q26

If \(C\) is the series whose \(n^\mathrm{th}\) term is \(\cos(nx)/n!\), and \(S\) the series \(\sin(nx)/n!\), calculate the sum from \(n = 1 \to \infty\) of \(C + iS\), and hence find the sum \(C\).

Strategy: Convert to the exponential form using \(e^{ix} = \cos(x) + i \sin(x)\), sum the terms then convert back to trig form and separate out the real part of the result.

Q27 Dielectric property of liquids

In the study of the dielectric properties of liquids and in electrochemical techniques that use potentiometry, the response of the solution to different electrical frequencies is studied. The general term for these experiments is Impedance Spectroscopy. In an experiment where a capacitor \(C\) and resistor \(R\) are in parallel, the impedance \(Z\), which is a complex quantity, is given by \(Z = (R^{1} + i\omega C)^{-1}\) where \(\omega\) is the frequency applied to the sample.

(a) Convert \(Z\) into the form \(Z = Z' - iZ''\).

(b) Plot \(Z''\) as ordinate, and \(Z'\) as abscissa. Show that the resulting curve is a semicircle. Use \(R = 5\) k\(\Omega\) and \(C = 1\, \mu\) F. Decide where high frequency is on the plot. This is not obvious from the graph because \(\omega\) is not on one of the axes.

Strategy: Multiply top and bottom of the expression by the complex conjugate. Look up the Matplotlib parametric method of plotting graphs.

Q28 Xray Diffraction. Scattering factor

Figure 7a shows two orthorhombic cells. On the right is a body centred cell. Calculate the scattering factor and intensity for diffraction this cell assuming that the two atoms comprising the unit cell are (a) different elements and (b) the same element. Assume the atom at the centre of the axes is at \((0,0,0)\).

Q29 Xray Diffraction. Friedel’s Law.

In a crystal that does not have a centre of symmetry the scattering factor,\(F\), eqn 21, for a plane \( h,k,l\) is different to that for \((-h,-k,-l)\). What are the intensities of the observed scattering in each case? What do you conclude from your result?

Strategy Convert the equations into cosine /sine form (see eqn. 16) and calculate the intensity. Recall that the scattering factor is squared to find the intensity.