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Math 551 Lecture Notes Complex Variables: a Very Short Practical Guide (Fundamentals and Contour Integrals)

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Math 551 Lecture Notes Complex Variables: a Very Short Practical Guide (Fundamentals and Contour Integrals) MATH 551 LECTURE NOTES COMPLEX VARIABLES: A VERY SHORT PRACTICAL GUIDE (FUNDAMENTALS AND CONTOUR INTEGRALS) Topics covered • Complex functions ◦ Complex differentiability; analytic functions ◦ Cauchy-Riemann equations (connection to harmonic functions) ◦ Multiple-valued functions and branch cuts • Preparing for contour integrals ◦ Poles and meromorphic functions ◦ Cauchy integral formula, Laurent series ◦ Contour integrals; contour deformation (in general) ◦ Residue theorem, calculating residues ◦ Trick for simple poles Introduction The main goal here is to introduce the fundamentals of complex analysis required to work with contour integrals that arise in the Fourier and Laplace transforms and the nice prop- erties of complex functions that inform the solutions. Our focus will be on these transform methods for solving PDEs, but it will be necessary to first learn the methods for computing contour integrals in general. As such, the treatment here is minimal and is by no means a thorough look at this rather deep subject. We will leave out some topics in favor of having more time to focus on the intended applications. Notation: The following notation will be used (refer back as needed): p • i = −1 (the imaginary unit) and z = x+iy is a complex number with real/imaginary parts x and y. A point z in the complex plane C is associated with (x; y) in R2. • A complex function f(z) with real/imaginary parts u and v can be written f(z) = f(x + iy) = u(x; y) + iv(x; y) • Γ is a contour in the complex plane, often parameterized by z(t) = (x(t); y(t)) R H • Γ f dz is a contour integral in the complex plane and Γ f dz is a contour integral over a closed contour (or `closed loop') 1. Review of complex numbers p • Complex numbers: C is the space of complex numbers z = x + iy where i = −1 is the `imaginary unit' and x = Re(z) = `real part'; y = Im(z) = `imaginary part': 1 2 COMPLEX VARIABLES Addition is component-wise as in R2 but i2 = −1 so z1z2 = (x1 + y1i)(x2 + y2i) = x1x2 − y1y2 + (y1x2 + x1y2)i: While tempting to compare to the plane R2, the `complex plane' has much more structure due to the multiplication of complex numbers. The conjugate (with an overbar) of z = x + iy and the modulus jzj (`magnitude') are p p z = x − yi; jzj = zz = x2 + y2: The conjugate distributes over sums/products (vw = vw etc.) and 1 1 Re(z) = (z + z); Im(z) = (z − z): 2 2 • Polar form: A complex number z can be written in rectangular or polar coordinates: z = x + iy or z = r cos θ + ir sin θ; r2 = x2 + y2; θ = arg z = tan−1(y=x) The values r and θ are the modulus and argument, respectively. Euler's formula and n-th roots: A critical result is Euler's formula, cos θ + i sin θ = eiθ (so z = reiθ in polar form) In particular, note that this implies that e2πi = 1 and zn = 1; n ≥ 1 an integer =) z = e2πik=n; k = 0; ··· n − 1 More generally, solving zn = c is done in polar form: zn = reiα =) z = r1=neiα/ne2πik=n; k = 0; ··· ; n − 1 iθ −iθ iθ1 iθ2 i(θ1+θ2) Other arithmetic is also easy in this form e.g. e = e and r1e · r2e = r1r2e : • Functions: A general complex function is a function of two variables, typically written as f(z; z) or a function of x; y for the input z = x + iy. It can be written in terms of real/imaginary components: f(z; z) = u(x; y) + iv(x; y) where u; v are real functions from the plane to R. Most real functions f(x) have `complexi- fied’ versions f(z) (just a function of z) e.g. f(x) = x2 =) f(z) = (x + iy)2 = x2 − y2 + i2xy: COMPLEX VARIABLES 3 2. Multi-valued functions, branch cuts A complex function can be multi-valued due to the formula e2πik = 1 for all integers k: Most functions are still single-valued dspite this, e.g. f(z) = z2 =) f(ze2πik) = z2e4πik = z2: However, multi-valued functions are also common. For example, consider f(z) = z1=2: Evaluating in polar coordinates with z = reiθ, f(reiθ) = f(reiθ · 1) = f(reiθ+i2πk) = r1=2eiθ=2eiπk; k = 0 or 1: That is, there are two versions of f that can be defined: ( f(z) = jzj1=2eiθ=2 (k = 0) f(z) = −|zj1=2eiθ=2 (k = 1) The two definitions are the branches of f. We must choose one branch to have a single val- ued function. The k = 0 version is called the principal branch of the function. Regardless of which branch is chosen, there is a jump across some line (segmaent) in the complex plane: The location of this line depends on the choice of argument. If we take θ 2 [−π; π] then there is a jump at θ = π since f(eiθ) ! i as θ increases to π ; f(eiθ) ! −i as θ decreases to −π so there is a jump at θ = π. The problem is that the argument θ = arg z is periodic , but ei arg z=2 has different values for θ's separated by 2π. We could also choose θ 2 [0; 2π], in which case the jump is across θ = 0 instead. The segment across which f is discontinuous is the branch cut and the endpoints are the branch points. In this case, we have one branch point at 0 and another `at 1' (at −∞ + 0i with θ 2 [−π; π] and 1 + 0i for θ 2 [0; 2π]. • Logarithms: An important example is the natural logarithm: ln(z) = ln(reiθe2πik) = ln(r) + iθ + 2πik = ln jf(z)j + i arg(z) + 2πik 4 COMPLEX VARIABLES with infinite number of branches (one for each integer k). The principal branch is ln(z) = ln(r) + iθ: This function has a branch cut where arg(z) jumps and a branch point at the origin, and discontinuous across this jump (by a value of 2πi). • Typically, the branch cut occurs where arg(z) jumps, and the range of θ determines the placement of the branch cut. For more complicated functions (to be discussed later), branch cuts can be segments or centered at branch points other than the origin. • Roots in general: Using the natural logarithm we may determine branch cuts for `n-th roots' and other powers using the definition zp = ep ln z: 1=m For example, consider (z − z1) : Using the above and the formula for ln z, 1 2πik (z − z )1=m = exp( ln(z − z )) = jz − z j1=m exp( arg(z − z )) 1 m 1 1 m 1 iα With z − z1 = re , this reads 1=m 1=m (z − z1) = r exp(2πikα=m): Each distinct value of exp(2πikα=m) yields a branch; there may be finite or infinitely many depending on the number of distinct values. Example (branch cut): Suppose we wish to define a branch of z4 so that f(z) = z4; f(1) = i; f is continuous along the real axis (except 0): There are four choices for z4, from the calculation f(ze2πik) = jzj1=4eiθ=4eπik=2: The one with f(1) = i is (plug in z = ei0) z4 = jzj1=4eiθ=4eiπ=2: The branch cut occurs at the endpoint of the range for θ = arg z: We needed continuity at θ = 0 and θ = π so the branch cut must be placed somewhere else e.g. by taking [−π=2; 3π=2] (any range including 0; π inside). This puts the branch cut at −π=2 . COMPLEX VARIABLES 5 3. Analytic functions A complex function f(z; z) has the form z + z z − z f(z; z) = u(x; y) + iv(x; y); x = ; y = : (3.1) 2 2 The function is called analytic or holomorphic if it is well-defined and solely a function of z. The `brute force' meaning of this is that if the expressions for x; y are plugged into (3.1), the z's cancel, leaving only z (so f = f(z)). This is equivalent to no dependence on z, written succinctly as @f f = f(z) () = 0: (3.2) @z The complex derivative of an analytic function is f(z + ∆z) − f(z) f 0(z) = lim : (3.3) ∆z!0 ∆z The limit can be taken over ∆z = ∆x + i∆y as ∆x; ∆y ! 0 in any way. Thus (3.3) is really two partial derivatives. However, when f is analytic, (3.3) is well-defined - the limit in any direction (∆x; ∆y ! 0 by any path) will give the same value f 0(z): To understand why f 0(z) is well defined, let z = x + iy, let f be analytic and write f(z) = u(x; y) + iv(x; y) We can take the derivative along the `real' or `imaginary' directions: 8 ∆x ! 0; ∆y = 0 (real) <> ∆z = ∆x + i∆y ! 0 with or :>∆x = 0; i∆y ! 0 (imaginary) This gives two values for f 0(z), which are f(z + ∆x) − f(z) @u @v f 0(z) = lim = + i ∆x!0 ∆x @x @x f(z + i∆y) − f(z) @u @v f 0(z) = lim = −i + ∆y!0 i∆y @y @y Equating the two, we obtain the Cauchy-Riemann equations, which also characterize analytic functions. Putting the results in a nice box: Analytic functions: For f = u(x; y) + iv(x; y), the following are equivalent: • f is well-defined and `a function of z only' (f = f(z) or equiv.
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