MF Ch 4

Morse & Feshbach Chapter 4 notes PhL 1.11.12 Chapter 4: Functions of a Complex Variable (348-492) 1 4.1 Complex Numbers and Variables (349)/ 1 4.2 Analytic Functions (356) 1 4.3 Derivatives of Analytic Functions, Taylor and Laurent Series (374) 1 4.4 Multivalued Functions (398) 2 4.5 Calculus of Residues: Gamma and Elliptic Functions (408) 2 4.6 Asymptotic Series; The Method of Steepest Descent (434) 2 4.7 Conformal Mapping (443) 2 4.8 Fourier Transforms (453) 2 Problems for Chapter 4 ( 50 problems!) 2 Tabulation of Properties of Functions of a Complex Variable (480) 2 Table of Special Function Data (486) 3 Bibliography 3 Chapter 4: Functions of a Complex Variable (348-492) Intro: M&F ask the question why complex is needed since physics is real, and their answers are good. This section is about scalar functions of a single complex variable. 4.1 Complex Numbers and Variables (349)/ Discussion of i as 90 degree rotation operator in the z plane. Rotation operator as first example and the de Moivre relation. Basics of complex variable math, including gradient. 2D electrostatics. Cauchy-Riemann equations for u and v as definition of analytic function. Contour integration. All just intro. 4.2 Analytic Functions (356) This section follows Ahlfors I would say. Derivative's role in definition of analytic function, then proof of the C-R equations. Conformal mapping idea, angles preserved, mapping in general 2D→2D. More detail on complex integration. Cauchy theorem says closed line integral gives 0 for analytic function. Hence allowed to deform a contour. Integration around a simple pole. Integration formulas for real and imaginary parts p 371, examples are Hilbert Transform p 372. Principle part integral above 4.2.17 and their notation is to put P in front of the integral. Then on to their favorite subject: impedances. The Poisson Formula tells you f(z) inside a circle on which you know f. More Hilbert Transform similar stuff. 4.3 Derivatives of Analytic Functions, Taylor and Laurent Series (374) f analytic => f' analytic. Taylor is the analytic thing, Laurent if you have some poles to worry about. Notion of essential singularity on page 380. Liouville's theorem (as used in WH technique!) p 381. Meromorphic function is analytic except for a finite number of poles (or multiple poles I assume). Power series and circle of convergence. Example is F(a,b,c,z). Analytic continuation on p 389, a subject I like. Uniqueness of the continuation. Branch points merely mentioned. Methods of doing analytic continuation. Power series stepping is pretty painful, not a preferred method. Some long examples here, not sure what they are. 4.4 Multivalued Functions (398) Here we have f(z) = z1/2 and we have the usual branch point and cut (they call it a branch line, not a cut at this point). Branches of a function. Riemann surfaces p 401 with attempt to draw 2-sheeted surface in the stereo pictures. Now they allow the word "cut" in the context of the sheets. They pick z(f) = th(f)/f and study it in detail, showing contours and so on. [ they do not use the word "sheet" ] 4.5 Calculus of Residues: Gamma and Elliptic Functions (408) Evaluation of integrals. Phrase "calculus of residues" just means deforming around the poles etc. Examples given where you evaluate integral by deforming around a cut. Can you invert a power series function (p 411)? A method is given using a complex integral. Notion of converting a series to an integral which I am quite familiar with. Integral representations. Gamma function defined by an integral representation. Product representation for gamma ( we have a little "special functions" section here fore the Gamma function). The dlnΓ/dz is written as ψ1(z) = Γ'/Γ. Duplication formula. Beta function p 425. Comment on series and product expansions for periodic functions p 426 top. What about functions with periods in both directions? (doubly periodic functions). We are then off on a discussion of the Jacobi elliptic functions which have this property, and the corresponding elliptic integrals which are the inverses of these functions. Theta functions appear here as well. 4.6 Asymptotic Series; The Method of Steepest Descent (434) The Method gives asymptotic series. So here is the M&F section on this interesting subject. 4.7 Conformal Mapping (443) Schwarz-Christoffel theorem as in Jackson lecture handout of Days of Olde. Use in electrostatics to study fringing fields. The Method of Inversion !!!! (short section) 4.8 Fourier Transforms (453) Their convention is that F(k) = 1/ ∫dx e+ikx f(x), the expo's sign agrees with Stak but not the constant in the projection. Validity of FT? Some discussion p 456 of Lebesgue integration ideas. The Fourier Integral Theorem. Parseval. Where is the FT analytic? Lots of strip stuff. Then the usual convolution integral which they call "the faltung" of two functions (folding). Diagonalization. Then on to the Poisson Sum Formula which I learned in Stak. Then the Laplace Transform. The Mellin Transform. Problems for Chapter 4 ( 50 problems!) 4.1 through 4.51 !! 50 problems! Tabulation of Properties of Functions of a Complex Variable (480) A little summary which includes some FT lookup tables on p 484, Schaum-like. A little table of Mellin transforms. Table of Special Function Data (486) Gamma related functions, the Jacobi elliptic functions, and Theta functions as well. Bibliography W&W. Ahlfors was 1953, same year as M&F and did not make the cut. Copson 1935 book. Watson on Bessel. Sneddon on FT's. Most of the references I have never heard of!