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extra_11 - m n power All other terms are higher in power...

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Extra Problems Sheet 11 Stephen Taylor May 25, 2005 1. For each of the following, determine whether the statement is true or false. Justify your answers with a brief explanation or counterexample. (a) If f is a nonconstant entire function, then | f ( z ) | is unbounded. This is the contrapositive of Louiville’s Theorem and is therefore true. (b) If f ( z ) has a zero of order m at z 0 , then f 0 ( z ) /f ( z ) has a simple pole at z 0 . This is a true statement. f has a Laurent series representation with smallest negative power m and f 0 has smallest negative power ( m - 1). When the two terms are divided, the result it a term in the form a j / ( z - a ) plus higher terms, which implies z 0 has the desired simple pole. (c) If f has a removable singularity at z 0 , then 1 /f ( z ) also has a removable singularity at z 0 . This is false and sin z/z is a counterexample. (d) If f and g have poles of order m and n respectively, at z 0 , then f ( z ) g ( z ) has a pole of order m + n at z 0 This is a true statement and is seen by multiplying the smallest negative power terms of the Laurent Series expansions of f and g to ﬁnd a new term of negative

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Unformatted text preview: m + n power. All other terms are higher in power than this one. (e) If f ( z ) has a pole of order m at z = 0, then f ( z 2 ) has a pole of order 2 m at z = 0. This is a true statement that follows directly from noting the Laurent Series representation of f and making the substitution z → z 2 . (f) If f has an essential singularity at z and g has a pole at z , then f + g has an essential singularity at z . This is a true statement and is obvious by summing the associated Laurent Series representations of both functions. (g) If f and g have an essential singularity at z , then f ( z ) /g ( z ) has a removable singularity at z . 1 This statement is false and a counterexample is f ( z ) = sin(1 /z ), g ( z ) = cos(1 /z ) which gives f/g = tan(1 /z ) which has an essential singularity at z . ± 2. Verify Picard’s Theorem for the function cos(1 /z ) at z = 0. 2...
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extra_11 - m n power All other terms are higher in power...

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