By R. J. Goult
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Extra resources for Applicable Mathematics: A Course for Scientists and Engineers
The complex variables z and ware related by the equation w = (2z + i)/(3 + z). Find in terms of x andy the real and imaginary parts of w. Hence find the locus of the point P representing z on the Argand diagram if (a) w is real, (b) w is purely imaginary. Solution. If z = x + iy, let w = u + iv. Then U . -'(3+x)2+y2 (a) v= 0 When w is real, x + 6y + 3 = 0. passing through the point -~i on the Argand diagram. (b) When w is purely imaginary, u = 0. i and radius y(37)/4. 24 APPLICABLE MATHEMATICS 2.
Neither the right-hand limit nor the left-hand limit of [J(x) as x tends to 0 is well defined. Hence, the function cannot be continuous from the right nor from the left. It is said to have a discontinuity of the second kind at x = 0. 1 Theorem 1 If, as x-+ a, f(x)-+ L 1 and g(x)-+ Lz, then (i) f(x) + g(x)-+ L1 + Lz. (ii) f(x)- g(x)-+ L1 - Lz, (iii) f(x) x g(x)-+ L 1 x Lz, and (iv) f(x)/g(x)-+ LdLz provided Lz =f 0. The four proofs are similar, although those for (iii) and (iv) are somewhat more complicated than those for (i) and (ii).
It is said to have a discontinuity of the second kind at x = 0. 1 Theorem 1 If, as x-+ a, f(x)-+ L 1 and g(x)-+ Lz, then (i) f(x) + g(x)-+ L1 + Lz. (ii) f(x)- g(x)-+ L1 - Lz, (iii) f(x) x g(x)-+ L 1 x Lz, and (iv) f(x)/g(x)-+ LdLz provided Lz =f 0. The four proofs are similar, although those for (iii) and (iv) are somewhat more complicated than those for (i) and (ii). We shall prove proposition (i) as a specimen. Given E > 0, we wish to find 8 such that I[f(x) + g(x)]- (L 1 + Lz)l < E for alllx- al < 8.
Applicable Mathematics: A Course for Scientists and Engineers by R. J. Goult