Applicable Mathematics: A Course for Scientists and by R. J. Goult

By R. J. Goult

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Additional info for Applicable Mathematics: A Course for Scientists and Engineers

Example text

We need some formal structure on which to base a decision as to whether or not this function should be regarded as continuous at x = 0. 3 Limits Definition 1. The functionf(x) is said to tend to the limit Las x tends to a if given any positive number €, there exists a corresponding number 5 such that lf(x)- Ll

Deduce that (zii) = (z)n. Hence prove that if z 0 is a complex root of the real polynomial equation a0 +a 1x + a 2x 2 + · · · + anxn = 0 then so also is z0 . 36 APPLICABLE MATHEMATICS 14. If z = eie show that cos(} and sin(} may be expressed in terms of z and z-1 ; More generally, show that cos n(J and sin n(J may be expressed in terms of zn and z-n. (cos 4(}- 4 cos 2(} + 3) and use this to evaluate * 1 for each integer 15. w is called a primitive nth root of 1 if wn = 1 and wm m in the range 0 < m < n (for example, i is a primitive 4th root of 1; -1 is a 4th root of 1 but not a primitive 4th root).

Is not bounded near x = 0 X we can neither say f(x) ~ +oo nor f(}c) f(x) 0 <0 and hence there is no neighbourhood of zero throughout whichf(x) >C. Consequently f(x) does not tend to infinity as x ~ 0. Similarly f(x) >- C for all x > 0 and sof(x) does not tend to minus infinity. 2 Arithmetic expressions involving unbounded functions The limiting behaviour of a simple arithmetic expression containing unbounded functions is not always simply related to the behaviour of the individual functions.

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