1550251515-Classical_Complex_Analysis__Gonzalez_

Elementary Functions 265 multiplicity. At any such pole (3 we have P((J) f:. 0, Q((J) = O, so R((J) = P((J) / Q((J) = oo. Hence ...

266 Chapter5 For w = (z^2 - 3z + 2)/(z^2 + 1), p = 2, so it has two zeros: 1, 2, and two poles: i, -i. The point oo is neither ...

Elementary Functions 267 Next, letting C = Bo-B~/4B 2 , (w-C)/B 2 = W,(+B 1 /2B 2 = Z, we get w = z^2 (5.16-3) an integral ratio ...

268 Chapter 5 5.17 THEJOUKOWSKI FUNCTION w = 1/2(z + 1/z) Letting z = rei^6 , w = u +iv, we have u +iv= ~ (reiB + ~ e-iB) which ...

Elementary Functions 269 and to the ray arg z = B+rr corresponds the same hyperbola. The semiaxes of the hyperbola are a= I cos ...

270 Chapter^5 y x Fig. 5.~!2 Exercises 5.3 1. Let ai, a 2 , ... , an be n given distinct complex numbers and /31, /32, ... , f3n ...

Elementary Functions ( ) z^3 - 6z^2 + llz - 6 cw=------- z3+1 271 Reduce w = (z^2 + z + 1)/(z - 2)^2 to the form W = Z^2 by sui ...

272 Chapter^5 real numbers a and b, with a^2 + b^2 f= 0, the polynomial af(x) + bg(x) has only real zeros. [J. Rainwater, Amer. ...

'Elementary Functions 273 e^2 krri = 1 ( k an integer) ezez' = ez+z' (additivity of exponents) lezl = ex,argez = y + 2k7r ez f: ...

274 Chapter^5 contained in S 0 corresponds in the w-plane the open ray L' = {w: w = exeim = bt,O < t < +oo} where ez = t a ...

Elementary Functions 275 the exponential also has a meaning when the exponent is any complex number, the following formulas, gen ...

276 Chapter^5 1 1 ' sechz = cschz = -- cosh z ' sinh z where coth z and csch z are not defined for z = krri, while tanh z, sech ...

Elementary Functions From (5.19-9) we obtain I sin zl^2 = sin^2 x cosh^2 y + cos^2 x sinh^2 y = sin^2 x(l + sinh^2 y) + (1 - sin ...

278 Chapter 5 y, s -1 so 81 82 i t >< t >< t ii II (') I (') _.._ z __ y_ =m z + 27T -· ------ ~ ---;:->----- ~ ...

Elementary Functions 279 0 < c < %7r, maps onto the lower part of the same branch, and as y increases from -oo to 0, that ...

280 Chapter^5 82 = {z: %rr < Rez < %rr,-oo < y < +oo} it maps exactly as So, and so on alternatively. Again we verif ...

Elementary Functions 281 is called a period strip for w = tan z. In particular, So= {z: -%rr < Rez::::;^1 / 2 rr,-oo < Imz ...

282 Chapter^5 lead to the impossibility 2 = 0. To find the images of the horizontal segments z = x + im, -%7r < x < (^1) / ...

Elementary Functions 283 Next, to determine the images of the vertical lines z = c + iy, 0 < !cl < %7r, -oo < y < +o ...

284 Chapter^5 Find the image of the square Q = {z: z = x + iy, 0 S x S %7r, 0 Sy S^1 / 2 7r} under w = ez. Find the image Q' ...