We introduce the Laurent Series as follows:

If f(z) is analytic inside and on the boundary of the ring-shaped region R (see diagram) bounded by two concentric circles C1 and C2 with center at a and respective radii r1 and r2 (r1 > r2) then for all z in R:

c

If f(z) is analytic inside and on the boundary of the ring-shaped region R (see diagram) bounded by two concentric circles C1 and C2 with center at a and respective radii r1 and r2 (r1 > r2) then for all z in R:

f(z) = å (z – a)

^{¥}_{n = -}_{¥}c_{n}(z – a)^{n}+ å^{¥}_{n =-}_{¥}_{ }c_{-n / }^{n}
where:

c

_{n}= 1/2 pi ò_{C}**1**^{ }f(w) dw /(w – a)^{n+1}n = 0,1, 2….._{- n }= 1/2 pi ò

_{C}**2**

^{ }f(w) dw /(w – a)

^{–n +1}n = -1,-2, -3…..

The other part of the series, consisting of negative powers, is called Taylor series.

__the principal part__. Either part or both may terminate or be identically zero. If the principal part is identically zero then f(z) is analytic at z = a since the derivative exists and the Laurent series is identical to the**:**

__More intricacies – and singularities__
Point z = a is called a zero or root of the function f(z) if f(a) = 0. If then f(z) is analytic at at z = a then the Taylor series:

f(z) = å

^{¥}_{n = 0}c**(z – a)**_{n}^{n}
must have c Taylor series – by calculating:

**= 0. If c**_{0}**¹ 0, the point a is called a simple zero (or a zero of order one). It could happen that c**_{1}**and perhaps several other next coefficients vanish. Then let c**_{1}**be the next vanishing coefficient (unless f(z) = 0) then the zero is said to be of order m. The order of a zero may be evaluated – without any knowledge of the**_{m}
lim

_{ z}**f(z)**_{® a}_{ }**/**(z – a)^{n}
For n = 1, 2, 3. The lowest value of n for which this limit doesn’t vanish is equal to the order of the zero.

If a function f(z) is analytic in the neighborhood of some point z = a with the exception of the point z = a itself then it is said to have an

**(or isolated singular point) at z = a. It’s customary to distinguish isolated singularities by the following types of behavior of f(z) as z ® a for an arbitrary function.***isolated singularity*Examples:

**÷**f(z)

**÷**

__<__B for a fixed B

**÷**f(z)

**÷**

**approaches infinity. Namely,**

**÷**f(z)

**÷**

**> M for**

**÷**z – a

**÷**

**< e**

3) Neither of the two cases above, in other words f(z) oscillates.

__Examples:__
1) f(z) = 1/ z – 1 (isolated singularity at z = 1)

a) If we demand Taylor expansion.

**÷**z**÷****< 1 we can obtain a**
b) ) If we seek: 1 <

**÷**z**÷**< 0 we obtain a Laurent expansion
2) f(z) = 1/ z (1 – z)

a) If 0 <

**÷**z

**÷**< 1 then we obtain a Laurent series

b) If 1 <

**÷**z**÷**< 0 we also obtain a Laurent expansion.__Problem for the Math Maven__

Consider the function: f(z) = 1/ (z+ 1) ((z + 3)

a) Find a Laurent series for: 1<

**÷**z**÷****< 3**
b) Find a Laurent series for 0 <

**÷**z + 1**÷****< 2**
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