ELEC3002-complex3

# ELEC3002-complex3 - ELEC3002/ELEC7005 Lecture 3...

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ITEE Complex 3/1 ELEC3002/ELEC7005 ELEC3002/ELEC7005 Lecture 3 (sections 1.5 and 1.6 in James) Singularities in complex analysis. Integration.

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ITEE Complex 3/2 Singularities, zeros and residues Singularities, zeros and residues A singularity is a point in the complex plane where the function f(z) ceases to be analytic e.g f(z) = 1/z . Normally, this means that the function is infinite at that point but it can also mean that there can be a choice of values e.g ( 29 2 2 f z z a = - A zero of f(z) is a point in the z-plane where f(z) = 0 . Now singularities can be classified in terms of the Laurent series expansion. Recall the expansion: ( 29 ( 29 ( 29 0 0 1 0 n n n n n n principal par t b f z a z z z z ( = = = - + - 1 4 2 4 3
ITEE Complex 3/3 A few options A few options If there is zero principal part about z = z 0 , then z 0 is a regular point. If f(z) has a finite number of terms in its principal part then f(z) has a singularity at z = z 0 called a pole . If there are m terms in the principal part , then the pole is said to be of order m . Another way of putting this last statement is to say that z 0 is a pole of order m if ( 29 ( 29 0 0 lim m z z z z f z b - - = see the Laurent series of the previous slide!

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ITEE Complex 3/4 If the principal part of the Laurent series has infinitely many terms then we have what is called is an essential singularity . Finally, if f(z) appears to be singular at z = z 0 , but it turns out that we can define a Power Series there, then we have a so called removable singularity . ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 1 3 1 pole of order 1 at z = 0 (simple pole) 1 pole of order 3 at z = 1 has an essential singularity at z = j sin has a removable singularity at z = 0 z j f z z f z z f z e f z z z - - - = = - = = Examples:
Complex 3/5 meromorphic functions meromorphic functions Functions whose only singularities are poles are called meromorphic and most engineering applications fall into this category. An interesting application of this is the expansion of transient responses from radar targets which are expressed as: 1 ( ) entire fn. n n s t n n f t a e = = + meromorphic fn when transformed into the freq domain ie has pole series. no poles

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ELEC3002-complex3 - ELEC3002/ELEC7005 Lecture 3...

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