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Since this holds for arbitrary ϵ > 0 , we see that f ˜ ( z ) = F ( z ) for all z c in B r ( c ) .

Finally, since

f ˜ ( c ) = lim z c f ˜ ( z ) = lim z c F ( z ) = F ( c ) ,

the equality of F and f ˜ on all of B r ( c ) is proved. This finishes the proof of part (1).

Prove part (2) of the preceding theorem.

Now, for the second kind of isolated singularity:

A complex number c is called a pole of a function f if there exists an r > 0 such that f is continuous on the punctured disk B ' ¯ r ( c ) , analytic at each point of B r ' ( c ) , the point c is not a removable singularity of f , and there exists a positive integer k such that the analytic function ( z - c ) k f ( z ) has a removable singularity at c .

A pole c of f is said to be of order n, if n is the smallest positive integer for which the function f ˜ ( z ) ( z - c ) n f ( z ) has a removable singularity at c .

  1. Let c be a pole of order n of a function f , and write f ˜ ( z ) = ( z - c ) n f ( z ) . Show that f ˜ is analytic on some disk B r ( c ) .
  2. Define f ( z ) = sin z / z 3 for all z 0 . Show that 0 is a pole of order 2 of f .

Let f be continuous on a punctured disk B ' ¯ r ( c ) , analytic at each point of B r ' ( c ) , and suppose that c is a pole of order n of f . Then

  1. For all z B r ' ( c ) ,
    f ( z ) = k = - n a k ( z - c ) k .
  2. The infinite series of part (1) converges uniformly on each compact subset K of B r ' ( c ) .
  3. For any piecewise smooth geometric set S B r ( c ) , whose boundary C S has finite length, and satisfying c S 0 ,
    C S f ( ζ ) d ζ = 2 π i a - 1 ,
    where A - 1 is the coefficient of ( z - c ) - 1 in the series of part (1).

For each z B r ' ( c ) , write f ˜ ( z ) = ( z - c ) n f ( z ) . Then, by [link] , f ˜ is analytic on B r ( c ) , whence

f ( z ) = f ˜ ( z ) ( z - c ) n = 1 ( z - c ) n k = 0 c k ( z - c ) k = k = - n a k ( z - c ) k ,

where a k = c n + k . This proves part (1).

We leave the proof of the uniform convergence of the series on each compact subset of B r ' ( c ) , i.e., the proof of part (2), to the exercises.

Part (3) follows from Cauchy's Theorem ( [link] ) and the computations in [link] . Thus:

C S f ( ζ ) d ζ = C r f ( ζ ) d ζ = C r k = - n a k ( z - c ) k d ζ = k = - n a k C r ( ζ - c ) k d ζ = a - 1 2 π i ,

as desired. The summation sign comes out of the integral because of the uniform convergence of the series on the compact circle C r .

  1. Complete the proof to part (2) of the preceding theorem. That is, show that the infinite series k = - n a k ( z - c ) k converges uniformly on each compact subset K of B r ' ( c ) . HINT: Use the fact that the Taylor series n = 0 c n ( z - c ) n for f ˜ converges uniformly on the entire disk B ¯ r ( c ) , and that if c is not in a compact subset K of B r ( c ) , then there exists a δ > 0 such that | z - c | > δ for all z K .
  2. Let f , c , and f ˜ be as in the preceding proof. Show that
    a - 1 = f ˜ ( n - 1 ) ( c ) ( n - 1 ) ! .
  3. Suppose g is a function defined on a punctured disk B r ' ( c ) that is given by the formula
    g ( z ) = k = - n a k ( z - c ) k
    for some positive integer n and for all z B r ' ( c ) . Suppose in addition that the coefficient a - n 0 . Show that c is a pole of order n of g .

Having defined two kinds of isolated singularities of a function f , the removable ones and the polls of finite order, there remain all the others, which we collect into a third type.

Let f be continuous on a punctured disk B ' ¯ r ( c ) , and analytic at each point of B r ' ( c ) . The point c is called an essential singularity of f if it is neither a removable singularity nor a poll of any finite order. Singularities that are either poles or essential singularities arecalled nonremovable singularities.

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Source:  OpenStax, Analysis of functions of a single variable. OpenStax CNX. Dec 11, 2010 Download for free at http://cnx.org/content/col11249/1.1
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