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A summary of taylor series functions and their properties, including some practice exercises relating the taylor series to the identity theorem.

Let S be a subset of C , let f : S C be a complex-valued function, and let c be a point of S . Then f is said to be expandable in a Taylor series around c with radius of convergence r if there exists an r > 0 such that B r ( c ) S , and f ( z ) is given by the formula

f ( z ) = n = 0 a n ( z - c ) n

for all z B r ( c ) .

Let S be a subset of R , let f : S R be a real-valued function on S , and let c be a point of S . Then f is said to be expandable in a Taylor series around c with radius of convergence r if there exists an r > 0 such that the interval ( c - r , c + r ) S , and f ( x ) is given by the formula

f ( x ) = n = 0 a n ( x - c ) n

for all x ( c - r , c + r ) .

Suppose S is an open subset of C . A function f : S C is called analytic on S if it is expandable in a Taylor series around every point c of S .

Suppose S is an open subset of R . A function f : S C is called real analytic on S if it is expandable in a Taylor series around every point c of S .

Suppose S is a subset of C , that f : S C is a complex-valued function and that c belongs to S . Assume that f is expandable in a Taylor series around c with radius of convergence r . Then f is continuous at each z B r ( c ) .

Suppose S is a subset of R , that f : S R is a real-valued function and that c belongs to S . Assume that f is expandable in a Taylor series around c with radius of convergence r . Then f is continuous at each x ( c - r , c + r ) .

If we let g be the power series function given by g ( z ) = a n z n , and T be the function defined by T ( z ) = z - c , then f ( z ) = g ( T ( z ) ) , and this theorem is a consequence of [link] and [link] .

Prove that f ( z ) = 1 / z is analytic on its domain.

HINT: Use r = | c | , and then use the infinite geometric series.

State and prove an Identity Theorem, analogous to [link] , for functions that are expandable in a Taylor series around a point c .

  1. Prove that every polynomial is expandable in a Taylor series around every point c . HINT: Use the binomial theorem.
  2. Is the exponential function expandable in a Taylor series around the number - 1 ?

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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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