Transcription of Notes on Complex Analysis in Physics
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Notes on Complex Analysis in Physics
Notes on Complex Analysis in PhysicsJim NapolitanoMarch 9, 2013 These Notes are meant to accompany a graduate level Physics course, to provide a basicintroduction to the necessary concepts in Complex Analysis . They are not complete, nor areany of the proofs considered rigorous. The immediate goal is to carry through enough of thework needed to explain the Cauchy Residue Theorem. Other material may be added Numbers and Complex FunctionsA Complex numberzcan be written asz=x+iyorz=rei withr 0wherei= 1, andx,y,r, and are real numbers. Clearly,x=rcos andy=rsin leading to a description in terms of the Complex plane. The Complex conjugate ofzisz =x iyorz =re i The modulus ofzis|z| z z=r= x2+y2and is often called the phase Complex functionf(z) typically returns a Complex number. Generically, we writef(z) =u(x,y) +iv(x,y)(1)for purposes of proofs or illustrations. The behavior of the (real) functionsu(x,y) andv(x,y)are critical for classifying Complex functions, as seen when we consider taking and AnalyticityWe define the derivativef (z) =df/dzof a Complex functionf(z) in the same was as we dofor the derivatives of real functions.
A complex function f(z) typically returns a complex number. Generically, we write ... applications in physics, and we will merely scratch the surface here. For example, ... where k, a, q, and >0 are all real variables. We use the second version above because this
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