Jan 20| Feb 1 | Mar 1 | Apr 5 | Phy 515 page
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1 Mar Look at the sketches of complex functions that had been assigned last time. Go over the one that is needed for completing the calculation just started, exp( i z2 ). Evaluate the integrals of cos( x2 ) and of sin( ). Show why the rotation of the contour makes sense, but onlyif you know what the integrand looks like in the complex plane.
Pick another function to integrate, and look at its behavior in the plane, the integral of sin( x ) / x along the real axis. Start picking apart the integrand as a function of z and have them sketch some results.
3 Mar
Describe the shape of the complex function tan( z ) in detail. Show how
to solve the homework problem involving the closed contour integral of
1/ ( z2 + z ) by pushing the contour to infinity and showing
that its constant value approaches zero.
Use the same technique to prove the fundamental theorem of algebra.
Spend some time first explaining why it's a non-trivial and important
theorem.
Complete the analysis of the integral from the day before,
sin( x ) / x. Do all the contour integrals.
A little discussion on the meaningof the word "integral."
5 Mar The Laurent series. Where does it converge? (In an annulus, if at all.) Can you differentiate it term by term? (Yes.) Can you do contour integrals term by term? (Yes.) From here you have the Residue Theorem.
Examine the doubly infinite series, S 1 / ( n2 + a2 ). Evaluate the two limiting cases, as a approaches zero and as a approaches infinity. Next, show how to do it exactly using the Watson-Sommerfeld Contour. Replace the sum by a contour integral, using the known properties of the cotangent function. Then evaluate the contour integral.