Complex Integration and Methods Prerequisites: The complex functions handout Concepts of primary interest: ∫ f() z dz = 0 for f(z) analytic in the interior Laplace property Poles of an analytic function Laurent series Residue theorem: fzdz()= 2π i Res(() fz ∫ ∑ m 1f () z dz Cauchy’s integral formula fa()= for f(z) analytic 2π i∫C za− Sample calculations: SC0: Generating a Laurent series SC3: Finding a residue SC5: Contour morphing to compute a sum SC6: Laplace property SC7: Standard contour integration SC9: Cauchy integral formula SC10: Integrals of trigonometric functions SC11: Integral representation of a special function SC12: Step function as a contour integral Applications Tools of the Trade Evaluating residues Integrals of real functions along the entire real axis: Closing contours with semicircular arcs in the upper and lower halfplane - Jordan's Lemma Integrals of real function over the positive real axis symmetry and pie wedges. Principal Value integrals Winding number Modified residue theorem *** Section not proofed. A basic knowledge of complex methods is crucial for graduate physics. This handout only illustrates a few of the standard methods, and the developments are not rigorous. For additional study, download the online text: Mathematical Tools for Physics by James Nearing from www.physics.miami.edu/nearing/mathmethods/. If you interest swells to the twenty dollar level, purchase the McGraw-Hill Schaum’s outline Complex Variables by Murray R. Spiegel. Next, purchase or check out a math-physics text such as Mathematics for Physicists by Susan Lea. If you find need to be a power applied user, move on to Functions of a Complex Variable by Carrier, Contact: [email protected] 5/30/2012 Krook and Pearson (McGraw-Hill 1966) after studying two of the previous suggestions. A less dated resource is Visual Complex Analysis by Tristan Needham. The Derivative of a Function of a Complex Variable: fz()21− fz () The Limit must be defined independent of the manner in which z1 and z2 approach zz, → z 12 o()zz21− zo. If z is represented as x + i y where x, y ∈ the condition requires that fx(0+∆ xy , 0 ) − fx (, 00 y ) fx (,00 y+∆ y ) − fx (,00 y ) Limit = Limit . The conclusion is that ∆→xy00 ∆∆x ∆→ iy (except for constant functions) differentiable functions of a complex variable must by complex valued. ∂∂ff Required: = −i [CI.1] ∂∂xy Analytic Functions of a complex variable: A function of a complex variable that is differentiable in a region is said to be analytic in that region. The function can be expanded about each point in the region as a Taylor’s series that has a non-zero radius of convergence. The Closed-Path Integral Property: The iy −∆x integral of complex function f(z) around any closed path bounding a closed region in which f(z) is analytic at every point is zero. This (,xy ) ∆ property of analytic functions is suggested by -i∆y i y examining the path integral of f(z) around a ∫ f() z dz small path. The argument and result are reminiscent of Stokes theorem. ∆x x fzdzfxy( )≈ ( , − ½ ∆ y ) ∆+ x fx ( + ½ ∆ xyiy , )( ∆ ) ∫ +fxy( , + ½ ∆ y )( −∆ x ) + fx ( − ½ ∆ xy , )( − iy ∆ ) 5/30/2012 Physics Handout Series.Tank: Complex Integration CI-2 ∂∂ff ∫ f() z dz≈ i +(i ∆∆ y x ) = 0 [CI.2] ∂∂yx(,)xy Cauchy’s integral Theorem: Property [CI.1] of differentiable functions was used to show that the integral vanishes. This result suggests that: ∫ f() z dz = 0 [CI.3] anypath bounding region of analyticity That is: the enclosed region in which the function is analytic is simply connected. The integral of complex function f(z) around any closed path bounding a closed region in which f(z) is analytic at every point is zero. The enclosed region must be simply connected; f(z) must be analytic everywhere in the entire enclosed region with no excluded points or patches. Exercise: List the new words that have been coined in this handout. Attempt to define them. The Laplace Property: It f(z) is represented as sum u(x,y) + i v(x,y) where u(x,y) and v(x,y) are real- valued functions, then the condition [CI.1] becomes the Cauchy-Riemann equations: ∂∂uv ∂ u ∂ v = and = − [CI.4] ∂∂xy ∂ y ∂ x Computing the second derivatives: ∂∂22uv ∂ 2 u ∂ 2 v ∂∂22uu = and = − ⇒ +=0 ∂x22 ∂∂ xy ∂ y ∂∂ yx ∂∂xy22 It follows that u(x,y) is a solution of the Laplace equation in two dimensions (as is v(x,y)). Considered as functions of the real variables x and y, the real and imaginary parts of an analytic function of a complex variable satisfy the Laplace equation in two dimensions. Exercise: Show the v(x,y) satisfies the 2D Laplace equation in the case that v(x,y) is the imaginary part of an analytic function of a complex variable. 5/30/2012 Physics Handout Series.Tank: Complex Integration CI-3 The closed-path integral and Laplace properties of analytic functions are enormously important. The integral property provides a technique to evaluate integrals that resist conventional techniques, and the Laplace property provides a tool for solving boundary value problems in two dimensions. Entire functions: A function that is analytic (has a representation as a convergent Taylor’s series) at every finite point in the complex plane is called an entire function. A function that is differentiable at every point will be analytic. A few examples of entire functions are 1, z2, ez and sin(z). Functions with poles and branch cuts -1 ½ Examples of functions that fail to be entire are (1 – z) , tan(π z), ln(z) and z . The function (1 - z)-1 cannot be expanded about or differentiated at z = 1. The function has a pole at the isolated point z = 1. The function tan(π z) has isolated poles for z equal to any integer. The story for ln(z) is a little more complicated. -1 Im(z) ln(z) = ln(|z|) + i tan ( /Re(z)) + i n 2 π The natural logarithm is not single-valued. Each z is mapped to an infinite set of values. As z tracks around a circular path centered on the origin, the imaginary part of ln(z) increases be smoothly and continuously by 2π. But, then you are back at the original z with a value of ln(z) that differs from the original by 2πi. A line must be drawn from the origin out to infinity. The function can be made continuous and differentiable everywhere in the complex plane except for points on that line. Such lines are called branch cuts and the endpoint of a branch cut is a branch point. Each such almost complete plane of values is a sheet of the function. The natural logarithm has an infinite number of sheets. The screw thread image above represents the imaginary part of the logarithm running to higher and higher values as the point z travels around a circle about the origin in the complex plane again and again. For z = |z| eiφ, ln[z] = ln[|z|] + iφ + i n2π. The value of φ is undefined at |z| = 0, and |z| = 0 is a branch point. The branch cut can run from the branch point to infinity along any non- intersecting path. The standard choices are the rays φ = 0 and φ = π. The constant φ-rays (relative to the branch point) are the useful cut choices for the natural logarithm. 5/30/2012 Physics Handout Series.Tank: Complex Integration CI-4 Natural Logarithm ln[|z|] ln[|z|] ln[|z|] + i π ln[|z|] + i π xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx ln[|z|] + i π ln[|z|] + i2π ln[|z|] - i π ln[|z|] Branch cut along the positive real axis. The Branch cut along the negative real axis. The labeled values are points just above and labeled values are points just above and below the real axis. The values progress below the real axis. The values progress continuously across the negative real axis continuously across the positive real axis but but jump by 2πi crossing the positive axis. jump by 2πi crossing the negative axis. The function z ½ has a branch cut. Consider the circle of points |z| = 1 and move around it in the CCW sense for z = ei φ, z ½ = ei φ/2. Start at z = -1; the square root increases smoothly from i and on to -1 are z moves around the circle t z = 1. Continuing, the square root approaches –i as the point z = -1 is approached. As the point z = -1 is approached, z ½ is about -i rather than i. The function is not continuous and therefore is not differentiable. The cut can run along any line from the origin to infinity, and a popular choice is a ray out along the negative imaginary axis. The new behavior is that after two trips around the circle, the value of z ½ does smoothly approach its initial value. Surface for Square Root: A strained rendition of the geometry of a function of two sheets with a branch cut. Note that, as the functions are complex valued, a faithful representation is not possible. The figure is to suggest that the values after one circuit fail to match, and that, after a second circuit, the values do match. 5/30/2012 Physics Handout Series.Tank: Complex Integration CI-5 Square Root ½ ½ ½ ½ i |z| |z| i |z| |z| xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx ½ ½ ½ i |z| - |z| – i |z|½ |z| Branch cut along the positive real axis. The Branch cut along the negative real axis. The labeled values are points just above and labeled values are points just above and below the real axis.