(a) i. Transform the Real integral into a complex integral\

\\\\int_0^(\\\\infty ) (x^(2))/((x^(2)+1)(x^(2)+4))dx

\ integral clearly specifying the appropriate path.\ 5 Marks\ ii. Locate and classify the poles and find the residue of each and\ hence evaluate the integral\ 10 Marks\ (b) i. Show that by using the substitution

z-e^(i\\\\theta )

\

I-\\\\int_0^(2\\\\pi ) (1)/(25-24cos\\\\theta )d\\\\theta

\ can be rewritten as\

(-1)/(i)o\\\\int_C (z)/(z(4z-3)(3z-4))dz

\ where

C

is the unit circle\ 5 Marks\ ii. Locate and classify the poles of this function and hence find the\ value of I\ 5 Marks\ (c) Using Complex Numbers show that\

(d)/(dz)cosz--sinz

\ 5 Marks\ (d) Give two examples of single valued functions and two examples of\ functions which are not single valued?\ 5 Marks

3. (a) i. Transform the Real integral into a complex integral 0(x2+1)(x2+4)x2dx integral clearly specifying the appropriate path. 5 Marks ii. Locate and classify the poles and find the residue of each and hence evaluate the integral 10 Marks (b) i. Show that by using the substitution zei I022524cos1d can be rewritten as i1Cz(4z3)(3z4)zdz where C is the unit circle 5 Marks ii. Locate and classify the poles of this function and hence find the value of I 5 Marks (c) Using Complex Numbers show that dzdcoszsinz 5 Marks (d) Give two examples of single valued functions and two examples of functions which are not single valued? 5 Marks