1. Show that the real part of \(i^{\log(1+i)}\) is \[e^{(4k+1)\pi^{2}/8 } \cos \{\tfrac{1}{4}(4k + 1)\pi\log 2\},\] where \(k\) is any integer.
2. If \(a\cos\theta + b\sin\theta + c = 0\), where \(a\), \(b\), \(c\) are real and \(c^{2} > a^{2} + b^{2}\), then \[\theta = m\pi + \alpha \pm i\log \frac{|c| + \sqrt{c^{2} – a^{2} – b^{2}}}{\sqrt{a^{2} + b^{2}}},\] where \(m\) is any odd or any even integer, according as \(c\) is positive or negative, and \(\alpha\) is an angle whose cosine and sine are \(a/\sqrt{a^{2} + b^{2}}\) and \(b/\sqrt{a^{2} + b^{2}}\).
3. Prove that if \(\theta\) is real and \(\sin\theta \sin\phi = 1\) then \[\phi = (k + \tfrac{1}{2})\pi pm i\log \cot \tfrac{1}{2}(k\pi + \theta),\] where \(k\) is any even or any odd integer, according as \(\sin\theta\) is positive or negative.
4. Show that if \(x\) is real then \[\begin{gathered} \frac{d}{dx} \exp\{(a + ib)x\} = (a + ib) \exp\{(a + ib) x\}, \\ \int \exp \{(a + ib)x\}\, dx = \frac{\exp{(a + ib)x}}{a + ib}.\end{gathered}\] Deduce the results of Ex. LXXXVII. 3.
5. Show that if \(a > 0\) then \(\int_{0}^{\infty} \exp\{-(a + ib)x\}\, dx = \frac{1}{a + ib}\), and deduce the results of Ex. LXXXVII. 5.
6. Show that if \((x/a)^{2} + (y/b)^{2} = 1\) is the equation of an ellipse, and \(f(x, y)\) denotes the terms of highest degree in the equation of any other algebraic curve, then the sum of the eccentric angles of the points of intersection of the ellipse and the curve differs by a multiple of \(2\pi\) from \[-i\{\log f(a, ib) – \log f(a, -ib)\}.\]
[The eccentric angles are given by \(f(a\cos\alpha, b\sin\alpha) + \dots = 0\) or by \[f\left\{\tfrac{1}{2} a \left(u + \frac{1}{u}\right),\ -\tfrac{1}{2} ib \left(u – \frac{1}{u}\right) \right\} + \dots = 0,\] where \(u = \exp i\alpha\); and \(\sum\alpha\) is equal to one of the values of \(-i\log P\), where \(P\) is the product of the roots of this equation.]
7. Determine the number and approximate positions of the roots of the equation \(\tan z = az\), where \(a\) is real.
[We know already (Ex. XVII. 4) that the equation has infinitely many real roots. Now let \(z = x + iy\), and equate real and imaginary parts. We obtain \[\sin 2x/(\cos 2x + \cosh 2y) = ax,\quad \sinh 2y/(\cos 2x + \cosh 2y) = ay,\] so that, unless \(x\) or \(y\) is zero, we have \[(\sin 2x)/2x = (\sinh 2y)/2y.\] This is impossible, the left-hand side being numerically less, and the right-hand side numerically greater than unity. Thus \(x = 0\) or \(y = 0\). If \(y = 0\) we come back to the real roots of the equation. If \(x = 0\) then \(\tanh y = ay\). It is easy to see that this equation has no real root other than zero if \(a \leq 0\) or \(a \geq 1\), and two such roots if \(0 < a < 1\). Thus there are two purely imaginary roots if \(0 < a < 1\); otherwise all the roots are real.]
8. The equation \(\tan z = az + b\), where \(a\) and \(b\) are real and \(b\) is not equal to zero, has no complex roots if \(a \leq 0\). If \(a > 0\) then the real parts of all the complex roots are numerically greater than \(|b/2a|\).
9. The equation \(\tan z = a/z\), where \(a\) is real, has no complex roots, but has two purely imaginary roots if \(a < 0\).
10. The equation \(\tan z = a\tanh cz\), where \(a\) and \(c\) are real, has an infinity of real and of purely imaginary roots, but no complex roots.
11. Show that if \(x\) is real then \[e^{ax} \cos bx = \sum_{0}^{\infty} \frac{x^{n}}{n!} \left\{ a^{n} – \binom{n}{2} a^{n-2} b^{2} + \binom{n}{4} a^{n-4} b^{4} – \dots \right\},\] where there are \(\frac{1}{2}(n + 1)\) or \(\frac{1}{2}(n + 2)\) terms inside the large brackets. Find a similar series for \(e^{ax} \sin bx\).
12. If \(n\phi(z, n) \to z\) as \(n \to \infty\), then \(\{1 + \phi(z, n)\}^{n} \to \exp z\).
13. If \(\phi(t)\) is a complex function of the real variable \(t\), then \[\frac{d}{dt} \log \phi(t) = \frac{\phi'(t)}{\phi(t)}.\]
[Use the formulae \[\phi = \psi + i\chi,\quad \log \phi = \tfrac{1}{2}\log(\psi^{2} + \chi^{2}) + i\arctan(\chi/\psi).]\]
14. Transformations. In Ch.III (Ex. XXI. 21 et seq., and Misc. Ex. 22 et seq.) we considered some simple examples of the geometrical relations between figures in the planes of two variables \(z\), \(Z\) connected by a relation \(z = f(Z)\). We shall now consider some cases in which the relation involves logarithmic, exponential, or circular functions.
Suppose firstly that \[z = \exp(\pi Z/a),\quad Z = (a/\pi) \log z\] where \(a\) is positive. To one value of \(Z\) corresponds one of \(z\), but to one of \(z\) infinitely many of \(Z\). If \(x\), \(y\), \(r\), \(\theta\) are the coordinates of \(z\) and \(X\), \(Y\), \(R\), \(\Theta\) those of \(Z\), we have the relations \[\begin{aligned} {2} x &= e^{\pi X/a} \cos(\pi Y/a),\qquad & y &= e^{\pi X/a} \sin(\pi Y/a),\\ X &= (a/\pi) \log r, & Y &= (a\theta/\pi) + 2ka,\end{aligned}\] where \(k\) is any integer. If we suppose that \(-\pi < \theta \leq \pi\), and that \(\log z\) has its principal value \(\log z\), then \(k = 0\), and \(Z\) is confined to a strip of its plane parallel to the axis \(OX\) and extending to a distance \(a\) from it on each side, one point of this strip corresponding to one of the whole \(z\)-plane, and conversely. By taking a value of \(\log z\) other than the principal value we obtain a similar relation between the \(z\)-plane and another strip of breadth \(2a\) in the \(Z\)-plane.
To the lines in the \(Z\)-plane for which \(X\) and \(Y\) are constant correspond the circles and radii vectores in the \(z\)-plane for which \(r\) and \(\theta\) are constant. To one of the latter lines corresponds the whole of a parallel to \(OX\), but to a circle for which \(r\) is constant corresponds only a part, of length \(2a\), of a parallel to \(OY\). To make \(Z\) describe the whole of the latter line we must make \(z\) move continually round and round the circle.
15. Show that to a straight line in the \(Z\)-plane corresponds an equiangular spiral in the \(z\)-plane.
16. Discuss similarly the transformation \(z = c\cosh(\pi Z/a)\), showing in particular that the whole \(z\)-plane corresponds to any one of an infinite number of strips in the \(Z\)-plane, each parallel to the axis \(OX\) and of breadth \(2a\). Show also that to the line \(X = X_{0}\) corresponds the ellipse \[\left\{\frac{x}{c\cosh(\pi X_{0}/a)}\right\}^{2} + \left\{\frac{y}{c\sinh(\pi X_{0}/a)}\right\}^{2} = 1,\] and that for different values of \(X_{0}\) these ellipses form a confocal system; and that the lines \(Y = Y_{0}\) correspond to the associated system of confocal hyperbolas. Trace the variation of \(z\) as \(Z\) describes the whole of a line \(X = X_{0}\) or \(Y = Y_{0}\). How does \(Z\) vary as \(z\) describes the degenerate ellipse and hyperbola formed by the segment between the foci of the confocal system and the remaining segments of the axis of \(x\)?
17. Verify that the results of Ex. 16 are in agreement with those of Ex. 14 and those of Ch. III, Misc. Ex. 25. [The transformation \(z = c\cosh(\pi Z/a)\) may be regarded as compounded from the transformations \[z = cz_{1},\quad z_{1} = \tfrac{1}{2}\{z_{2} + (1/z_{2})\},\quad z_{2} = \exp(\pi Z/a).]\]
18. Discuss similarly the transformation \(z = c\tanh(\pi Z/a)\), showing that to the lines \(X = X_{0}\) correspond the coaxal circles \[\{x – c\coth(2\pi X_{0}/a)\}^{2} + y^{2} = c^{2}\operatorname{cosech}^{2}(2\pi X_{0}/a),\] and to the lines \(Y = Y_{0}\) the orthogonal system of coaxal circles.
19. The Stereographic and Mercator’s Projections. The points of a unit sphere whose centre is the origin are projected from the south pole (whose coordinates are \(0\), \(0\), \(-1\)) on to the tangent plane at the north pole. The coordinates of a point on the sphere are \(\xi\), \(\eta\), \(\zeta\), and Cartesian axes \(OX\), \(OY\) are taken on the tangent plane, parallel to the axes of \(\xi\) and \(\eta\). Show that the coordinates of the projection of the point are \[x = 2\xi/(1 + \zeta),\quad y = 2\eta/(1 + \zeta),\] and that \(x + iy = 2\tan \frac{1}{2}\theta \operatorname{Cis}\phi\), where \(\phi\) is the longitude (measured from the plane \(\eta = 0\)) and \(\theta\) the north polar distance of the point on the sphere.
This projection gives a map of the sphere on the tangent plane, generally known as the Stereographic Projection. If now we introduce a new complex variable \[Z = X + iY = -i\log \tfrac{1}{2}z = -i\log \tfrac{1}{2}(x + iy)\] so that \(X = \phi\), \(Y = \log \cot \frac{1}{2}\theta\), we obtain another map in the plane of \(Z\), usually called Mercator’s Projection. In this map parallels of latitude and longitude are represented by straight lines parallel to the axes of \(X\) and \(Y\) respectively.
20. Discuss the transformation given by the equation \[z = \log \left(\frac{Z – a}{Z – b}\right),\] showing that the straight lines for which \(x\) and \(y\) are constant correspond to two orthogonal systems of coaxal circles in the \(Z\)-plane.
21. Discuss the transformation \[z = \log \left\{\frac{\sqrt{Z – a} + \sqrt{Z – b}}{\sqrt{b – a}}\right\},\] showing that the straight lines for which \(x\) and \(y\) are constant correspond to sets of confocal ellipses and hyperbolas whose foci are the points \(Z = a\) and \(Z = b\).
[We have \[\begin{aligned} {2} \sqrt{Z – a} + \sqrt{Z – b} &= \sqrt{b – a}\, \exp(& &x + iy), \\ \sqrt{Z – a} – \sqrt{Z – b} &= \sqrt{b – a}\, \exp(&-&x – iy);\end{aligned}\] and it will be found that \[|Z – a| + |Z – b| = |b – a|\cosh 2x,\quad |Z – a| – |Z – b| = |b – a|\cos 2y.]\]
22. The transformation \(z = Z^{i}\). If \(z = Z^{i}\), where the imaginary power has its principal value, we have \[\exp(\log r + i\theta) = z = \exp(i\log Z) = \exp(i\log R – \Theta),\] so that \(\log r = -\Theta\), \(\theta = \log R + 2k\pi\), where \(k\) is an integer. As all values of \(k\) give the same point \(z\), we shall suppose that \(k = 0\), so that \[\begin{equation*} \log r = -\Theta,\quad \theta = \log R. \tag{1} \end{equation*}\]
The whole plane of \(Z\) is covered when \(R\) varies through all positive values and \(\Theta\) from \(-\pi\) to \(\pi\): then \(r\) has the range \(\exp(-\pi)\) to \(\exp\pi\) and \(\theta\) ranges through all real values. Thus the \(Z\)-plane corresponds to the ring bounded by the circles \(r = \exp(-\pi)\), \(r = \exp\pi\); but this ring is covered infinitely often. If however \(\theta\) is allowed to vary only between \(-\pi\) and \(\pi\), so that the ring is covered only once, then \(R\) can vary only from \(\exp(-\pi)\) to \(\exp \pi\), so that the variation of \(Z\) is restricted to a ring similar in all respects to that within which \(z\) varies. Each ring, moreover, must be regarded as having a barrier along the negative real axis which \(z\) (or \(Z\)) must not cross, as its amplitude must not transgress the limits \(-\pi\) and \(\pi\).
We thus obtain a correspondence between two rings, given by the pair of equations \[z = Z^{i},\quad Z = z^{-i},\] where each power has its principal value. To circles whose centre is the origin in one plane correspond straight lines through the origin in the other.
23. Trace the variation of \(z\) when \(Z\), starting at the point \(\exp \pi\), moves round the larger circle in the positive direction to the point \(-\exp \pi\), along the barrier, round the smaller circle in the negative direction, back along the barrier, and round the remainder of the larger circle to its original position.
24. Suppose each plane to be divided up into an infinite series of rings by circles of radii \[\dots,\quad e^{-(2n+1)\pi},\ \dots,\quad e^{-\pi},\quad e^{\pi},\quad e^{3\pi},\ \dots,\quad e^{(2n+1)\pi},\ \dots.\] Show how to make any ring in one plane correspond to any ring in the other, by taking suitable values of the powers in the equations \(z = Z^{i}\), \(Z = z^{-i}\).
25. If \(z = Z^{i}\), any value of the power being taken, and \(Z\) moves along an equiangular spiral whose pole is the origin in its plane, then \(z\) moves along an equiangular spiral whose pole is the origin in its plane.
26. How does \(Z = z^{ai}\), where \(a\) is real, behave as \(z\) approaches the origin along the real axis? [\(Z\) moves round and round a circle whose centre is the origin (the unit circle if \(z^{ai}\) has its principal value), and the real and imaginary parts of \(Z\) both oscillate finitely.]
27. Discuss the same question for \(Z = z^{a+bi}\), where \(a\) and \(b\) are any real numbers.
28. Show that the region of convergence of a series of the type \(\sum\limits_{-\infty}^{\infty} a_{n}z^{nai}\), where \(a\) is real, is an angle, a region bounded by inequalities of the type \(\theta_{0} < \operatorname{am} z < \theta_{1}\) [The angle may reduce to a line, or cover the whole plane.]
29. Level Curves. If \(f(z)\) is a function of the complex variable \(z\), we call the curves for which \(|f(z)|\) is constant the level curves of \(f(z)\). Sketch the forms of the level curves of \[\begin{aligned} {2} z – a \quad& \text{(concentric circles)}, \qquad& (z – a)(z – b) \quad& \text{(Cartesian ovals)}, \\ (z – a)/(z – b) \quad& \text{(coaxal circles)}, \qquad& \exp z \quad& \text{(straight lines)}.\end{aligned}\]
30. Sketch the forms of the level curves of \((z – a)(z – b)(z – c)\), \((1 + z\sqrt{3} + z^{2})/z\). [Some of the level curves of the latter function are drawn in Fig. 59, the curves marked i–vii corresponding to the values \[.10,\quad 2 – \sqrt{3} = .27,\quad .40,\quad 1.00,\quad 2.00,\quad 2 + \sqrt{3} = 3.73,\quad 4.53\] of \(|f(z)|\). The reader will probably find but little difficulty in arriving at a general idea of the forms of the level curves of any given rational function; but to enter into details would carry us into the general theory of functions of a complex variable.]
31. Sketch the forms of the level curves of (i) \(z\exp z\), (ii) \(\sin z\). [See Fig. 60, which represents the level curves of \(\sin z\). The curves marked i–viii correspond to \(k = .35\), \(.50\), \(.71\), \(1.00\), \(1.41\), \(2.00\), \(2.83\), \(4.00\).]
32. Sketch the forms of the level curves of \(\exp z – c\), where \(c\) is a real constant. [Fig. 61 shows the level curves of \(|\exp z – 1|\), the curves i–vii corresponding to the values of \(k\) given by \(\log k = -1.00\), \(-.20\), \(-.05\), \(0.00\), \(.05\), \(.20\), \(1.00\).]
33. The level curves of \(\sin z – c\), where \(c\) is a positive constant, are sketched in Figs. 62, 63. [The nature of the curves differs according as to whether \(c < 1\) or \(c > 1\). In Fig. 62 we have taken \(c = .5\), and the curves i–viii correspond to \(k = .29\), \(.37\), \(.50\), \(.87\), \(1.50\), \(2.60\), \(4.50\), \(7.79\). In Fig. 63 we have taken \(c = 2\), and the curves i–vii correspond to \(k = .58\), \(1.00\), \(1.73\), \(3.00\), \(5.20\), \(9.00\), \(15.59\). If \(c = 1\) then the curves are the same as those of Fig. 60, except that the origin and scale are different.]
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34. Prove that if \(0 < \theta < \pi\) then \[\begin{aligned} {3} \cos\theta &+ \tfrac{1}{3} \cos 3\theta &&+ \tfrac{1}{5} \cos 5\theta &&+ \dots = \tfrac{1}{4} \log \cot^{2}\tfrac{1}{2}\theta,\\ \sin\theta &+ \tfrac{1}{3} \sin 3\theta &&+ \tfrac{1}{5} \sin 5\theta &&+ \dots = \tfrac{1}{4}\pi,\end{aligned}\] and determine the sums of the series for all other values of \(\theta\) for which they are convergent. [Use the equation \[z + \tfrac{1}{3}z^{3} + \tfrac{1}{5}z^{5} + \dots = \tfrac{1}{2} \log \left(\frac{1 + z}{1 – z}\right)\] where \(z = \cos\theta + i\sin\theta\). When \(\theta\) is increased by \(\pi\) the sum of each series simply changes its sign. It follows that the first formula holds for all values of \(\theta\) save multiples of \(\pi\) (for which the series diverges), while the sum of the second series is \(\frac{1}{4}\pi\) if \(2k\pi < \theta < (2k + 1)\pi\), \(-\frac{1}{4}\pi\) if \((2k + 1)\pi < \theta < (2k + 2)\pi\), and \(0\) if \(\theta\) is a multiple of \(\pi\).]
35. Prove that if \(0 < \theta < \frac{1}{2}\pi\) then \[\begin{aligned} {3} \cos\theta &- \tfrac{1}{3} \cos 3\theta &&+ \tfrac{1}{5} \cos 5\theta &&- \dots = \tfrac{1}{4}\pi,\\ \sin\theta &- \tfrac{1}{3} \sin 3\theta &&+ \tfrac{1}{5} \sin 5\theta &&- \dots = \tfrac{1}{4} \log (\sec\theta + \tan\theta)^{2};\end{aligned}\] and determine the sums of the series for all other values of \(\theta\) for which they are convergent.
36. Prove that \[\cos\theta \cos\alpha + \tfrac{1}{2} \cos 2\theta \cos 2\alpha + \tfrac{1}{3} \cos 3\theta \cos 3\alpha + \dots = -\tfrac{1}{4} \log \{4(\cos\theta – \cos\alpha)^{2}\},\] unless \(\theta – \alpha\) or \(\theta + \alpha\) is a multiple of \(2\pi\).
37. Prove that if neither \(a\) nor \(b\) is real then \[\int_{0}^{\infty} \frac{dx}{(x – a)(x – b)} = -\frac{\log(-a) – \log(-b)}{a – b},\] each logarithm having its principal value. Verify the result when \(a = ci\), \(b = -ci\), where \(c\) is positive. Discuss also the cases in which \(a\) or \(b\) or both are real and negative.
38. Prove that if \(\alpha\) and \(\beta\) are real, and \(\beta > 0\), then \[\int_{0}^{\infty} \frac{d}{x^{2} – (\alpha + i\beta)^{2}} = \frac{\pi i}{2(\alpha + i\beta)}.\] What is the value of the integral when \(\beta < 0\)?
39. Prove that, if the roots of \(Ax^{2} + 2Bx + C = 0\) have their imaginary parts of opposite signs, then \[\int_{-\infty}^{\infty} \frac{dx}{Ax^{2} + 2Bx + C} = \frac{\pi i}{\sqrt{B^{2} – AC}},\] the sign of \(\sqrt{B^{2} – AC}\) being so chosen that the real part of \(\{\sqrt{B^{2} – AC}\}/Ai\) is positive.
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