FURMAN MTH 350 - Lecture 36: Examples of Taylor Series (3 pages)

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Lecture 36: Examples of Taylor Series

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Lecture 36: Examples of Taylor Series

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Furman University
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Mth 350 - Complex Variables
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Lecture 36 Examples of Taylor Series Dan Sloughter Furman University Mathematics 39 May 6 2004 36 1 Examples of Taylor series Example 36 1 Let f z ez Then f is entire and so its Maclaurin series will converge for all z in the plane Now f n 0 e0 1 for n 0 1 2 3 and so X zn z2 z3 z e 1 z n 2 3 n 0 for all z C Example 36 2 It follows from the previous example that 2z e X 2z n n n 0 X 2n n 0 n zn for all z C Later we will prove the uniqueness of power series representations from which it will follow that the expression above is the Maclaurin series for e2z Example 36 3 Similarly iz e n X i n 0 and e iz n zn X 1 n in n n 0 1 zn Hence iz iz e e X 1 1 n in n n 0 X 2i2n 1 2n 1 X 2i 1 n 2n 1 z z z 2n 1 2n 1 n 0 n 0 n Thus for all z C eiz e iz X 1 n z 2n 1 z3 z5 z7 z sin z 2i 2n 1 3 5 7 n 0 Example 36 4 We will see later that we may differentiate a power series as we would a polynomial that is term by term From this it will follow that cos z X 1 n 2n 1 z 2n n 0 2n 1 X 1 n z 2n 2n n 0 z z2 z4 z6 2 4 6 for all z C Example 36 5 We now have sinh z i sin iz i X 1 n i2n 1 z 2n 1 n 0 2n 1 X 1 2n z 2n 1 n 0 X n 0 z 2n 1 z 2n 1 2n 1 z3 z5 z7 3 5 7 and cosh z cos iz X z 2n z2 z4 z6 1 2n 2 4 6 n 0 for all z C Example 36 6 We have seen previously that X 1 zn 1 z z2 z3 1 z n 0 2 when z 1 Hence X 1 z 2n 1 z 2 z 4 1 z2 n 0 and X 1 1 n z 2n 1 z 2 z 4 z 6 1 z2 n 0 for all z with z 1 Example 36 7 For another example using the geometric series 1 1 z 1 1 z n X 1 z n n 0 X 1 n z 1 n n 0 1 z 1 z 1 2 z 1 3 for all z with z 1 1 Example 36 8 We have 1 1 1 1 1 2 4 6 1 z z z 2 1 z2 z4 2 4 2 2 2 z z z 1 z z z for all z with 0 z 1 Note

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