Q1 page 196 Q5Q2 page 197 Q 8 and 9Q3 page 206 Q 7Q4 page 220 Q3 and 5Q5 page 239 Q2(a,c,d)Q6 ...

Question

Q1  page 196 Q5Q2  page 197 Q 8 and 9Q3  page 206 Q 7Q4  page 220 Q3 and 5Q5  page 239 Q2(a,c,d)Q6  page 243 Q1Q7  page 248  Q3Q8  page 267  Q1,2,8Q9  page 290  Q 1,2

Robert · Accepted Answer

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.

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

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