ELEC3002-complex4

ELEC3002-complex4 - ELEC3002/ELEC7005 Lecture 4 Residue...

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ITEE Complex 4/1 ELEC3002/ELEC7005 ELEC3002/ELEC7005 Lecture 4 Residue Calculus (Section 1.64- end in James)
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ITEE Complex 4/2 What’s it good for? What’s it good for? Well, the coefficients of b k are interesting, since the first one ( 29 1 1 2 C b f z dz j π = is a multiple of the contour integral around a closed contour inside the appropriate annulus and containing z 0 in its interior. The coefficient b 1 is called the residue of f(z) at z 0 and is typically found, not by integrating , but by other devices.
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ITEE Complex 4/3 How does it work? How does it work? The purpose of the residue integration is to evaluate integrals ( 29 C f z dz b b Smart Idea! If f(z) is analytic everywhere on and in C, then result equals zero. if f(z) has a singularity at a point z = z 0 , but is otherwise analytic on and in C then f(z) has a Laurent expansion: ( 29 ( 29 ( 29 1 2 0 2 0 0 0 .... n n n b b f z a z z z z z z b = = - + + + - -
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ITEE Complex 4/4 Now the coefficient of the first term in the principal part of the series is ( 29 1 1 2 C b f z dz j π = In other words: ( 29 1 2 c f z dz jb = So, if we can find b 1 easily, we can evaluate the integral! The ( 29 0 1 z z b res f z = = here the integration is only around the residue (anticlockwise)
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ITEE Complex 4/5 So all that remains is to find the formula for the residue. Example: for a simple pole ( 29 ( 29 ( 29 ( 29 ( 29 2 1 0 1 0 2 0 0 0 1 0 0 1 0 ( ) ...... ( ) ... b f z a a z z a z z z z z z f z b z z a a z z = + + - + - + - - = + - + - + Now let z z 0 , then the RHS approaches b 1 and this gives a formula. ( 29 0 0 1 0 ( ) lim ( ) z z z z res f z b z z f z b = = = -
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ITEE Complex 4/6 Another simpler formula for the residue at a simple pole can be used if we start from ( f(z ) has a simple pole at z 0 ) ( 29 ( 29 ( ) p z f z q z = A Taylor series of q(z) with centre at z 0 can be written ( 29 ( 29 ( 29 ( 29 ( 29 2 0 0 0 0 ... 2! z z q z z z q z q z - = - + + Substitute this into f=p/q and then f into the previous formula for a simple residue we get: ( 29 ( 29 ( 29 ( 29 0 0 0 0 ( ) z z z z p z p z res f z res q z q z = = = =
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ITEE Complex 4/7 Formula for a residue at a Pole of any Formula for a residue at a Pole of any order order See the derivation in Kreysig page 783. It basically is a
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This note was uploaded on 07/07/2009 for the course ELEC 3002 at Queensland.

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ELEC3002-complex4 - ELEC3002/ELEC7005 Lecture 4 Residue...

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