3 2 1 1 π π i ii iii iv v vi vii fig 61 k 29 37 50

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- 3 - 2 - 1 1 - π π I II III IV V VI VII Fig. 61. 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.] - π 2 π 2 3 π 2 0 π I IV V VI VII VIII II III Fig. 62. - π 2 π 2 3 π 2 0 I IV V VI VII II III Fig. 63.
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[X : 237] EXPONENTIAL, AND CIRCULAR FUNCTIONS 543 34. Prove that if 0 < θ < π then cos θ + 1 3 cos 3 θ + 1 5 cos 5 θ + · · · = 1 4 log cot 2 1 2 θ, sin θ + 1 3 sin 3 θ + 1 5 sin 5 θ + · · · = 1 4 π, and determine the sums of the series for all other values of θ for which they are convergent. [Use the equation z + 1 3 z 3 + 1 5 z 5 + · · · = 1 2 log 1 + z 1 - z where z = cos θ + i sin θ . When θ is increased by π the sum of each series simply changes its sign. It follows that the first formula holds for all values of θ save multiples of π (for which the series diverges), while the sum of the second series is 1 4 π if 2 kπ < θ < (2 k + 1) π , - 1 4 π if (2 k + 1) π < θ < (2 k + 2) π , and 0 if θ is a multiple of π .] 35. Prove that if 0 < θ < 1 2 π then cos θ - 1 3 cos 3 θ + 1 5 cos 5 θ - · · · = 1 4 π, sin θ - 1 3 sin 3 θ + 1 5 sin 5 θ - · · · = 1 4 log(sec θ + tan θ ) 2 ; and determine the sums of the series for all other values of θ for which they are convergent. 36. Prove that cos θ cos α + 1 2 cos 2 θ cos 2 α + 1 3 cos 3 θ cos 3 α + · · · = - 1 4 log { 4(cos θ - cos α ) 2 } , unless θ - α or θ + α is a multiple of 2 π . 37. Prove that if neither a nor b is real then Z 0 dx ( x - a )( x - b ) = - 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 α and β are real, and β > 0, then Z 0 d x 2 - ( α + ) 2 = πi 2( α + ) .
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[X : 237] THE GENERAL THEORY OF THE LOGARITHMIC, 544 What is the value of the integral when β < 0? 39. Prove that, if the roots of Ax 2 +2 Bx + C = 0 have their imaginary parts of opposite signs, then Z -∞ dx Ax 2 + 2 Bx + C = πi B 2 - AC , the sign of B 2 - AC being so chosen that the real part of { B 2 - AC } /Ai is positive.
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APPENDIX I (To Chapters III, IV, V) The Proof that every Equation has a Root Let Z = P ( z ) = α 0 z n + α 1 z n - 1 + · · · + α n be a polynomial in z , with real or complex coefficients. We can represent the values of z and Z by points in two planes, which we may call the z - plane and the Z -plane respectively. It is evident that if z describes a closed path γ in the z -plane, then Z describes a corresponding closed path Γ in the Z -plane. We shall assume for the present that the path Γ does not pass through the origin. To any value of Z correspond an infinity of values of am Z , differing by multiples of 2 π , and each of these values varies continuously as Z de- scribes Γ. * We can select a particular value of am Z corresponding to each point of Γ, by first selecting a particular value corresponding to the initial γ z Fig. A.
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