1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

7.3 • LAURENT SERIES REPRESENTATIONS 275

9. Find two Laurent series for z-^1 (4 - z) -^2 involving powers of z and state where

they are valid.

10. Find t hree Laurent series for (z^2 - 5z + 6)-^1 centered at a= 0.

1 1. Find the Laurent series for Log ( ~::) , where a and b are positive real numbers
with b > a > 1, and state where the series is valid. Hint: For these conditions,

show that Log(~=:) = Log(l - i) - Log(l - n.

12. 
    

    Can Log z be represented by a Maclaurin series or a Laurent series about the point
    a = O? Explain your answer.

    
    


13. 
    

    Use the Maclaurin series for sin z and then long division to get the Laurent series
    for csc z with a = O.
    QQ
    1 4. Show that cosh (z + ~) = I:; a,..z", where the coefficients can be expressed in
    n=-oo
    t he form a,,= 2 ~ J:" cosnOcosh(2cosO)d0. Hint: Let the path of integration be
    the circle ct ( 0).

    
    


14. 
    

    Consider the real-valued function u (8) = 5 _ 4 ';,.,. 6.

    
    

(a) Use the substitution cos8 = ~ (z + ±) and obtain

• z 1 1 1 l
  u(^9 ) = f (z) = (z- 2)(2z- 1) = 3 1 - ~ - 31 - 2z·

(b) Expand the function f (z) in part (a) in a Laurent series that is valid

in the annulus A(O,! , 2).

(c) Use the substitutions cos (n8) = ~ (z" + z- ") in part (b) and obtain

QQ

the Fourier series for 1J (0): u (0) = k + k I: rn+I cos (n9).

n = l

16. The Bessel function J,. (z) is sometimes defined by the generating function

exp[~(t-D] = "~QQJ,..(z)t".

Use the circle ct (0) as the contour of integration and show that

J., (z) = -11" cos (nO -zsin 0) dO.

7r 0

00

17. Suppose that the Laurent expansion f (z) = I:; a,,z" converges in the annulus

n=-oo

A(O,r1,r2), where r1 < 1 < r2. Consider the real-valued function u(8) = f (e'^9 )

and show that u ( 8) has the Fourier series expansion

00

u(O) = I (eie) = L a,..e•nB,

n=-oo

where