Math Methods for Polymer Physics Lecture 1: Series Representations of Functions Series analysis is an essential tool in polymer physics and physical sci- ences, in general. Though other broadly speaking, a series expansion allows one to analyze an arbitrarily complicated function into the sum of a simpler set of functions. Though other series expansions exist, two are especially useful: the Taylor series and Fourier series. In a crude way, we may think of both series as 2 different ways of approximating, or “fitting”, a given function to a simpler form. For further reading on Taylor series and Fourier series see chapters 5 and 14, respectively, of Arken and Weber's text, Mathematical Methods for Physicists. 1 Taylor Series Let's start with Taylor series expansions. The Taylor expansion is a repre- sentation of a function, say f(x), as an infinite power series in the polynomi- n als, (x−x0) , where x0 is some reference point for the independent variable, x. Why are Taylor series useful? Well, let's say you have a complicated function: x3 f(x) = ln cos x2 + 2 + : (1) 3 This function is plotted in Fig. 1. Often it's sufficient and useful to have a simpler description of the func- tion in the neighborhood of some point, say x = x0. For many func- tions, you may replace f(x) with a power series expansion in polynomials of ∆x = x − x0 distance from the reference point. 1 X n f(x) = an (∆x) n=0 2 3 = a0 + a1∆x + a2 (∆x) + a3 (∆x) + ::: (2) What are these coefficients an? The first one can be deduced from x = x0 and ∆x = 0, so that 2 3 f(x0) = a0 + a1∆x + a2 (∆x) + a3 (∆x) + ::: = a0 (3) | {z } 0 1 Figure 1: Plot of f(x) in eq. (1), dark solid line. For many applications, it is often necessary to know behavior of the functionf(x) near some point, say x0 = 2. The Taylor series expansions for f(x) around x = x0 including 1, 2, 3, and 4 terms only are shown as labelled. To find the higher-order (larger n) coefficients, take derivatives of both sides. Note that after this operation the right side is still a Taylor series. 0 2 f (x0) = a1 + 2a2∆x + 3a3 (∆x) + ::: = a1 (4) | {z } 0 In general, we may show 1 dnf a = : (5) n n! dxn x=x0 In order to find the Taylor series expansion we need only to take derivatives of f(x) evaluated only at the point of reference, x = x0. Eqs. (2) and (5) define the Taylor series expansion of a functions of a single variable. Functions which can be represented by a Taylor series are known as analytic functions. Notice from eq. (2) that as x ! x0 and ∆x ! 0 the higher order (large n) terms in the power series expansion go to zero very quickly. Hence, if one is interested in f(x) sufficiently close to x0, a Taylor series expansion truncated to include only a few leading terms may often be sufficient to approximate the function. Geometrically, we can think of this in terms of a \local" description of a function near x = x0. 2 3 0 (∆x) 00 (∆x) 000 f(x) = f(x0) + ∆xf (x0) + f (x0) + f (x0) + ::: (6) | {z } | {z } 2! 3! constant linear | {z } | {z } parabolic cubic 2 This shows that sufficiently close a point of interest that analytic functions are well approximated by constant plus a sloped, linear correction plus a parabolic correction plus :::. Further away from a given reference point at x = x0, the less and less a function looks like a straight line. In order to get a better a approximation, you need functions with more wiggles (e.g. higher order polynomials). Let's try some examples. Example 1: Expand ln(x) in a Taylor series around x0 = 1. a0 = ln 1 = 0 d 1 a = ln(x) = = 1 1 dx x x=1 x=1 d2 d 1 1 a = ln(x) = = − = −1 2 dx2 dx x x2 x=1 x=1 x=1 d3 2 a = ln(x) = = 2 3 dx3 x3 x=1 x=1 n In general, an = (−1) (n − 1)! for n > 1. (x − 1)2 (x − 1)3 (x − 1)4 ln(x) = (x − 1) − + − + ::: 2 3 4 1 X (−1)n(x − 1)n = (7) n n=1 1 Example 2: Expand in a Taylor around x = 0. 1 − x a0 = 0 d 1 1 a = = = 1! 1 dx 1 − x (1 − x)2 x=0 x=0 d2 1 2 a = = = 2! 2 dx2 1 − x (1 − x)3 x=0 x=0 d3 1 1 × 2 × 3 a = = = 3! 3 dx3 1 − x (1 − x)4 x=0 x=0 In general, an = n!. So 1 1 X = xn (8) 1 − x n=0 3 which is the well-known geometric series. These particular series, eqs. (7) and (8), do not converge for all values of x. When the series does not converge, for some large enough ∆x, the n successive terms terms an(∆x) become larger than that the sum of the previous terms, meaning that adding more terms in the series expansion does not provide a better approximation, and the Taylor series fails to rep- 1 resent the function. For ln(x) around x0 = 1 and 1 − x around x0 = 0, these only converge for j∆xj < 1. In general we may define Rc as the radius of convergence of the Taylor P1 n series of f(x) around x = x0. If j∆xj < Rc, then n=0 an(∆x) = f(x). Otherwise series does not provide a good approximation of f(x) (adding more terms makes things worse). There are some functions for which Rc ! 1 and the Taylor series always converges. Important examples include ex, sin x, cos x. These functions arise in many contexts, so it is useful to commit these series to memory. Example 3: Expand ex around x = 0. Well, first notice n d x x e = e = 1 dxn x=0 x=0 From eq. (5) this gives right away the Taylor series coefficient of ex 1 x2 x3 x4 X xn ex = 1 + x + + + + ::: = : (9) 2! 3! 4! n! n=0 The Taylor series representation of ex is a particularly useful way to see that d x x P1 xn dx (e ) = e . Indeed, it is reasonable to view n=0 n! as the definition of ex. You should also commit expressions of sin x and cos x to memory. These converge for all x: x3 x5 x7 sin x = x − + − + ::: (10) 3! 5! 7! x2 x4 x6 cos x = 1 − + − + ::: (11) 2! 4! 6! Notice that these expansions allow you to derive the following important identity, eix = cos x + i sin x; (12) which is used heavily in Fourier analysis. It is reasonably straightforward to generalize the Taylor series expansion for a function of a single variable to a mutli-variable function, say f(x; y), 4 expanded around the point x = x0 and y = y0: @f @f f(x; y) = f(x ; y ) + ∆x + ∆y 0 0 @x @y 1 h @2f @2f @2f i + (∆x)2 + 2∆x∆y + (∆y)2 2! @x2 @x@y @y2 1 h @3f @3f + (∆x)3 + 3(∆x)2∆y 3! @x3 @x2@y @3f @3f i +3∆x(∆y)2 + (∆y)3 + ::: (13) @x@y2 @y3 where ∆x = x − x0, ∆y = y − y0 and all partial derivatives are evaluated at (x0; y0). This expansion can be confirmed by taking first, second, third (etc.) derivatives of both sides of the equation above. Why is the Taylor expansion a useful description? In many physical systems, the full expression for a function may be impossible to write down (i.e. PE of strongly interacting mixtures of charged particles). But often, equilibrium and dynamic behavior depends only on local properties of func- tion. By \local", we mean, sufficiently close to some set of values for the independent variable. As a concrete example, consider a colloidal bead in a laser trap (Fig. 2) , an experimental tool which has been exploited to measure the forces generated by single macromolecules. If the bead has a polarizability, α, then when it is subject to an electric field, E, it obtains a dipole moment, p = αE. The potential energy of a polarized object in an electric field 1 is simply, U = − 2 p · E, while the energy required to polarize the bead is 2 Upolarization = jpj =(2α). Therefore, if the polarizable bead is subject to an electric field E(x) that varies in space (as near the focal point of a laser beam, the net electrostatic interaction between the bead and the field is described by the potential energy, α U(x) = − jE(x)j2: (14) 2 Hence, the potential energy is lowest in regions where the electric-field in- tensity, jE(x)j2, is highest. This explains why a small polarizable object, like colloidal beads, are drawn into the focal point of a high-intensity laser (shown schematically in Fig. 2). In general, the pattern of electric field intensity, jE(x)j2, may be rather complicated. But, if we are interested only in the behavior very close to the center of the trap, the behavior always has the same simple form, U2 2 U(∆x) = −U0 + U1∆x + (∆x) + :::: (15) |{z} | {z } 2 constant =0 | {z } quadratic 5 Figure 2: Top: a schematic depiction of a polarizable bead near to the high- intensity focal point of a laser beam. Bottom: A sketch of U, the potential energy of a optically-trapped colloidal particle in terms of ∆x, the deviation from the center of the trap.