Miscellaneous examples on chapter x 1 show that the

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MISCELLANEOUS EXAMPLES ON CHAPTER X. 1. Show that the real part of i log(1+ i ) is e (4 k +1) π 2 / 8 cos { 1 4 (4 k + 1) π log 2 } , where k is any integer. 2. If a cos θ + b sin θ + c = 0, where a , b , c are real and c 2 > a 2 + b 2 , then θ = + α ± i log | c | + c 2 - a 2 - b 2 a 2 + b 2 , where m is any odd or any even integer, according as c is positive or negative, and α is an angle whose cosine and sine are a/ a 2 + b 2 and b/ a 2 + b 2 . 3. Prove that if θ is real and sin θ sin φ = 1 then φ = ( k + 1 2 ) π ± i log cot 1 2 ( + θ ) , where k is any even or any odd integer, according as sin θ is positive or negative. 4. Show that if x is real then d dx exp { ( a + ib ) x } = ( a + ib ) exp { ( a + ib ) x } , Z exp { ( a + ib ) x } dx = exp ( a + ib ) x a + ib .
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[X : 237] EXPONENTIAL, AND CIRCULAR FUNCTIONS 535 Deduce the results of Ex. lxxxvii . 3. 5. Show that if a > 0 then Z 0 exp {- ( a + ib ) x } dx = 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 π from - i { log f ( a, ib ) - log f ( a, - ib ) } . [The eccentric angles are given by f ( a cos α, b sin α ) + · · · = 0 or by f 1 2 a u + 1 u , - 1 2 ib u - 1 u + · · · = 0 , where u = exp ; and α 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 2 x/ (cos 2 x + cosh 2 y ) = ax, sinh 2 y/ (cos 2 x + cosh 2 y ) = ay, so that, unless x or y is zero, we have (sin 2 x ) / 2 x = (sinh 2 y ) / 2 y. 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 5 0 or a = 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 5 0. If a > 0 then the real parts of all the complex roots are numerically greater than | b/ 2 a | .
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[X : 237] THE GENERAL THEORY OF THE LOGARITHMIC, 536 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 = X 0 x n n ! a n - n 2 a n - 2 b 2 + n 4 a n - 4 b 4 - . . . , where there are 1 2 ( n + 1) or 1 2 ( n + 2) terms inside the large brackets. Find a similar series for e ax sin bx .
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