Taylor Series

The Taylor series is particularly useful for analyzing how functions behave in limiting cases. This is essential when translating a mathematical expression into physical intuition, and I would argue that the ability to do such translations is one of the most important skills an aspiring physicist needs. Below I give a dozen examples from Intermediate Physics for Medicine and Biology, selected to give you practice with Taylor series. In each case, expand the function in the dimensionless variable that I specify. For every example—and this is crucial—interpret the result physically. Think of this blog post as providing a giant homework problem about Taylor series.

Find the Taylor series of:

1. Eq. 2.26 as a function of bt (this is Problem 26 in Chapter 2). The function is the solution for decay plus input at a constant rate. You will need to look up the Taylor series for an exponential, either in Appendix D or in your favorite math handbook. I suspect you’ll find this example easy.
2. Eq. 4.69 as a function of ξ (this is Problem 47 in Chapter 4). Again, the Taylor series for an exponential is required, but this function—which arises when analyzing drift and diffusion—is more difficult than the last one. You’ll need to use the first four terms of the Taylor expansion.
3. The argument of the inverse sine function in the equation for C(r,z) in Problem 34 of Chapter 4, as a function of z/a (assume r is less than a). This expression arises when calculating the concentration during diffusion from a circular disk. Use your Taylor expansion to show that the concentration is uniform on the disk surface (z = 0). This calculation may be difficult, as it involves two different Taylor series. 
4. Eq. 5.26 as a function of ax. Like the first problem, this one is not difficult and merely requires expanding the exponential. However, there are two equations to analyze, arising from the study of countercurrent transport. 
5. Eq. 6.10 as a function of z/c (assume c is less than b). You will need to look up or calculate the Taylor series for the inverse tangent function. This expression indicates the electric field near a rectangular sheet of charge. For z = 0 the electric field is constant, just as it is for an infinite sheet.
6. Eq. 6.75b as a function of b/a. This equation gives the length constant for a myelinated nerve axon with outer radius b and inner radius a. You will need the Taylor series for ln(1+x). The first term of your expansion should be the same as Eq. 6.75a: the length constant for an unmyelinated nerve with radius a and membrane thickness b.
7. The third displayed equation of Problem 46 in Chapter 7 as a function of t/t_C. This expression is for the strength-duration curve when exciting a neuron. Interestingly, the short-duration behavior is not the same as for the Lapicque strength-duration curve, which is the first displayed equation of Problem 46.
8. Eq. 9.5 as a function of [M']/[K]. Sometimes it is tricky to even see how to express the function in terms of the required dimensionless variable. In this case, divide both sides of Eq. 9.5, to get [K']/[K] in terms of [M']/[K]. This problem arises from analysis of Donnan equilibrium, when a membrane is permeable to potassium and chloride ions but not to large charged molecules represented by M’.
9. The expression inside the brackets in Eq. 12.42 as a function of ξ. The first thing to do is to find the Taylor expansion of sinc(ξ), which is equal to sin(ξ)/ξ. This function arises when solving tomography problems using filtered back projection.
10. Eq. 13.39 as a function of a/z. The problem is a little confusing, because you want the limit of large (not small) z, so that a/z goes to zero. The goal is to show that the intensity falls off as 1/z² for an ultrasonic wave in the Fraunhoffer zone.
11. Eq. 14.33 as a function of λk_BT/hc. This problem really is to determine how the blackbody radiation function behaves as a function of wavelength λ, for short wavelength (high energy) photons. You are showing that Planck’s blackbody function does not suffer from the ultraviolet catastrophe.
12. Eq. 15.18 as a function of x. (This is Problem 15 in Chapter 15). This function describes how the Compton cross section depends on photon energy. Good luck! (You’ll need it).

Brook Taylor

Who was Taylor? Brook Taylor (1685-1731) was an English mathematician and a fellow of the Royal Society. He was a champion of Newton’s version of the calculus over Leibniz’s, and he disputed with Johann Bernoulli. He published a book on mathematics in 1715 that contained his series.