Math 3228, Semester 2, 2003

Complex Analysis for Applications

(co-taught with James McCoy and Alan McIntosh)

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Lecture notes:

• Week 7: Winding numbers; argument principle; Rouche's theorem; stability of ODE.[pdf]
• Week 8: Conformal mapping; Mobius transformations; Gamma function. [pdf]
[Errata: in the first bullet on page 7, "a=d=1 and b=c=0" should be "a=d=0 and b=c=1". At the end of the proof in Theorem 11, z_0 should be f(z_0), and on the next line, f'(z_0)=0 should be f'(z_0) \neq 0. Thanks to Andrew Solomon for these corrections.]
• Week 9: Riemann-Zeta function; prime number theorem; functional equation. [pdf] [Erratum: on page 4, the poles of f are at the multiples of 2 pi i, not at the integers. Thanks to Aditha Guha Roy for the correction.]

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Problem sheets and assignments:

• Assignment 3 [pdf] and solution [pdf]

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Java applets

Applet 1: The complex plane Applet 2: Elementary complex maps Applet 3: Möbius transforms Applet 4: Multi-valued functions Applet 5: The complex derivative

Applet 6: The complex integral Applet 7: Laurent series Applet 8: The fundamental theorem of Calculus Applet 9: The residue theorem Applet 10: The argument principle

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Miscellaneous links:

• Do imaginary numbers actually exist?
• The sci.math FAQ on why e^{\pi i} = -1.
• An M.C. Escher print depicting reflection by a sphere (kind of a 3D analogue of the inversion map).

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