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**Extra info for Applicable Mathematics: A Course for Scientists and Engineers**

**Sample text**

Find the regions of the complex plane for which l-z+a- <1 ' z -a~ = 1, or >I where a is a fixed complex number whose real part is positive. 18. Find the equation of the circle described in the complex plane on the join of the points a, b as diameter. 19. Find the equation of the circle through the points 1, i, 1 + i, and determine its centre and radius. 20. (i) Prove that an equation for a line passingthrough fixed points z 1 and z 2 is given by arg {(z - z 1)/(z 2 - z 1)} = 0. (ii) Prove that an equation for a circle passing through fixed points z h z 2, and z 3 is given by 21.

41 FUNCTIONS OF A SINGLE REAL VARIABLE With this formal definition of limit it is easy to confirm that the function defined by f(x) = x for all rational values of x, f(x) = 0 for all irrational values of x, does tend to zero as x tends to zero. In fact, if e is any positive number then for every value of x such that lxl

Most modern texts on analysis contain an account of the essential features of the theory; for the reader who wishes to pursue the matter in more detail there is a comprehensive and reasonably accessible treatment to be found in E. ). 3. The number e A detailed discussion of the exponential function, exp x, is given in section 3. 1. From the initial definition of the function as a power series x2 x3 xn exp x = 1 + x +-+- + · · · +- + · · · 2! 3! n! we obtain the number e as the value of that function when x = 1 1 1 1 e =I+ 1 +- +- + · · · +- + · · · 2!