Q1 page 196 Q5 Q2 page 197 Q 8 and 9 Q3 page 206 Q 7 Q4 page 220 Q3 and 5 Q5 page 239 Q2(a,c,d) Q6 page 243 Q1 Q7 page 248 Q3 Q8 page 267 Q1,2,8 Q9 page 290 Q 1,2

1. Re derive the Maclaurin series (3) in Sec. 59 for the function f (z) = cos z by
(a) using the definition




In Sec. 34 and appealing to the Maclaurin series (1) for in Sec. 59
Sol:
From the Maclaurin series of in sec.59 and for | | ,
∑
( )
Replace z with –z, we get,
∑( )
( )
Using the definition of cosz and for | | ,







(∑
( ) ( ) ( )
)
{
∑
( )
∑
( )
( )
∑( )
( )
( )
∑( )
( )
( )
(b) showing that
( ) ( ) ( ) ( )
Sol:
( ) ∑
( )
( )

( )


( )


( ) ( )




(
( )


( )

)
( ) (
( )


( )

) ( )
Any number of times we differentiate this expression, the result stays the same.
This is because of the fact that we are getting the original expression (multiplied by a constant) back
every time we differentiate it twice.
Hence,
( ) (
( )


( )

)
( ) (
( )


( )

)
Hence,
( ) ( )
( )
2. With the aid of the identity (see Sec. 34)
(


)
expand cos z into a Taylor series about the point z0 = π/2.
Sol:
From the taylor series expansion, we can say that, | |
( ) ( )
( )

( )
( )

( )

for,
( )
( )


( )

( )
( )

( )





Replacing z in the above expression with (


),
(


) (

)
(


)



(


)



Hence,
(


) (

)
(


)



(


)



∑( )
(

)

( )
3. Use the identity sinh(z + πi) = −sinh z, verified in Exercise 7(a), Sec. 35, and the fact that sinh z is
periodic with period 2πi to find the Taylor series for sinh z about the point z0 = πi....