Extrema of the Sinc Function

In Intermediate Physics for Medicine and Biology, Russ Hobbie and I write

The function sin(x)/x has its maximum value of 1 at x = 0. It is also called the sinc(x) function.

Sinc(x) oscillates like sin(x), but its amplitude decays as 1/x. If sin(x) is zero then sinc(x) is also zero, except for the special point x = 0, where 0/0 becomes 1.

A plot of the sinc function.

Trigonometric Delights, by Eli Maor

In IPMB, Russ and I don’t evaluate the values of x corresponding to local maximum and minimum values of sinc(x). Eli Maor examines the peak values of f(x) = sinc(x) in his book Trigonometric Delights. He writes

We now wish to locate the extreme points of f(x)—the points where it assumes its maximum or minimum values. And here a surprise is awaiting us. We know that the extreme points of g(x) = sinx occur at all odd multiples of π/2, that is, at x = (2n+1)π/2. So we might expect the same to be true for the extreme points of f(x) = (sinx)/x. This, however, is not the case. To find the extreme point, we differentiate f(x) using the quotient rule and equate the result to zero:

          f’(x) = (x cosx – sinx)/x² = 0.         (1)

Now if a ratio is equal to zero, then the numerator itself must equal to zero, so we have x cosx – sinx = 0, from which we get

          tan x = x.                                         (2)

Equation (2) cannot be solved by a closed formula in the same manner as, say, a quadratic equation can; it is a transcendental equation whose roots can be found graphically as the points of intersection of the graphs of y = x and y = tan x.

A plot of y=tanx versus x and y=x versus x.

The extreme values are at x = 0, 4.49 = 1.43π, 7.73 = 2.46π, etc. As x becomes large, the roots approach (2n+1)π/2.

Eli Maor is a rare breed: a writer of mathematics. Russ and I cite his wonderful book e, The Story of a Number in Chapter 2 of IPMB. I also enjoyed The Pythagorean Theorem: A 4,000-year History. Maor has written many books about math and science. His most recent came out in May: Music by the Numbers--From Pythagoras to Schoenberg. I put it on my summer reading list.

A Dozen Units from Intermediate Physics for Medicine and Biology

Medical and biological physics have their share of colorful and sometimes obsolete units. For the most part, Intermediate Physics for Medicine and Biology sticks with standard metric, or SI, units; mass, distance and time are in kilograms, meters, and seconds (mks). Some combinations of units are given special names, usually in honor of a famous physicist, such as the newton (N) for kg m s^-2. I have always found the units for electricity and magnetism difficult to remember. The coulomb (C) for charge is easy enough, but units such as the tesla (T) for magnetic field strength in kg s^-1 C^-1 are tricky. IPMB uses some common non-SI units, such as the liter (l) for 10^-3 m³, the angstrom (Å) for 10^-10 m, and the electron volt (eV) for 1.6 × 10^-19 J.

Let’s count down a dozen unfamiliar units discussed in Intermediate Physics for Medicine and Biology. We’ll start with the least important, and end with the one you really need to know.

12. The roentgen (R). Chapter 16 of IPMB states that the roentgen “is an old unit of [radiation] exposure equivalent to the production of 2.58 × 10^-4 C kg^-1 in dry air.” The unit’s name written out as “roentgen” begins with a lower case letter “r” even though Wilhelm Roentgen’s last name starts with an upper case “R.” It's always that way with units.

11. The diopter (diopter). The diopter is a nickname for m^-1, just as the hertz is a nickname for s^-1. It is used mainly when discussing the power, or vergence, of a lens, and appears in Chapter 14 of IPMB. The diopter does not have a symbol, you just write out “diopter” (“dioptre” if you are English, but that is so wrong).

10. The einstein (E). Homework Problem 2 of Chapter 14 defines the einstein as “1 mol of photons.” Units like the mole (mol) and the einstein are really dimensionless numbers: a mole is 6 × 10²³ molecules and an einstein is 6 × 10²³ photons. John Wikswo and I have proposed the leibniz (Lz) to be 6 × 10²³ differential equations. Some define the einstein as the energy of a mole of photons, so be careful when using this unit. I’ll let you guess who the unit was named for.

9. The poise (P). Chapter 1 of IPMB analyzes the coefficient of viscosity, which is often expressed in units of poise or centipoise. The poise is a leftover from the old centimeter-gram-second system of units, and is equal to a gram per centimeter per second. The viscosity of water at 20 °C is about 1 cP. The poise is named after Jean Leonard Marie Poiseuille (sort of), just as the unit of capacitance (the farad) is kind of named after Micheal Faraday. The mks unit of viscosity is the poiseuille (Pl), where 1 Pl = 10 P. The poiseuille is not used much, probably because no one can pronounce it.

8. The torr (Torr). Pressure is measured in many units. The torr is nearly the same as a millimeter of mercury (mmHg), and is named after the Italian physicist Evangelista Torricelli. The SI unit for pressure is the pascal (Pa), a nickname for a newton per square meter. One Torr is about 133 Pa. The bar (bar) is 100,000 Pa, and is approximately equal to one atmosphere (atm). How confusing! All five units—torr, bar, atm, mmHg, and pascal—are used often, so you need to know them all.

7. The barn (b). The barn measures area and is 10^-28 m². It is equivalent to 100 fm² (the femtometer is also known as a fermi). Nuclear cross sections are measured in barns. By nuclear physics standards a barn is a pretty big cross section. The term barn comes from the idiom about “hitting the broad side of a barn.”

6. The debye (D). Homework Problem 3 in Chapter 6 of IPMB introduces the debye. It is defined as 10^-18 statcoulomb cm, where a statcoulomb is the unit of charge in the old cgs system. It is equivalent to 3.34 × 10^-30 C m. The debye is named after Dutch physicist Peter Debye, and measures dipole moment. The dipole moment of a water molecule is 1.85 D.

5. The candela (cd). Radiometry measures radiant energy using SI units. Photometry measures the sensation of human vision with its own oddball collection of units, such as lumens, candelas, lux, and nits. A candela depends on the color of the light; for green 1 cd is equal to a radiant intensity of about 0.0015 watts per steradian. A burning candle has a luminous intensity of about 1 cd.

4. The svedberg (Sv). The centrifuge is a common instrument in biological physics. A particle has a sedimentation coefficient equal to its sedimentation velocity per unit of centrifugal acceleration. The units of speed (m s^-1) divided by acceleration (m s^-2) is seconds, so sedimentation coefficient has dimensions of time. The svedberg is equal to 10^-13 s. IPMB gives the symbol as “Sv”, but sometimes it is just “S” (easily confused with a unit of conductance called the siemens and a unit of effective dose called the sievert). The unit is named after the Swedish chemist Theodor Svedberg, who invented the ultracentrifuge.

3. The curie (Ci). The curie is an older unit of radioactivity that is now out of fashion. It is named in honor of Pierre and Marie Curie, and it measures the activity, equal to the disintegration rate. The SI unit for activity is the becquerel (Bq), or disintegrations per second. The becquerel is named after Henri Becquerel, the French physicist who discovered radioactivity. One curie is 3.7 × 10¹⁰ Bq. The cumulated activity is the total number of disintegrations, and is a dimensionless number often expressed in Bq s (why bother?). An older unit for cumulated activity is the odd-sounding microcurie hour (µCi h).

2. The Hounsfield unit (HU). The Hounsfield unit is used to measure the x-ray attenuation coefficient µ during computed tomography. It is a dimensionless quantity defined by Eq. 16.25 in IPMB: H = 1000 (µ – µ_water)/µ_water (for some reason Russ Hobbie and I use H rather than HU). The unit is strange because everyone says the attenuation coefficient is so many Hounsfield units, including the word “units” (you never say a force is so many “newton units”). The attenuation coefficient of water is 0 HU. Air has a very small small attenuation coefficient, so on the Hounsfield scale it is -1000 HU. Many soft tissues have an attenuation coefficient on the order of +40 HU, and bone can be more than +1000 HU. The unit is named after English electrical engineer Godfrey Hounsfield, who won the 1979 Nobel Prize in Physiology or Medicine for developing the first clinical computed tomography machine.

and the winner is....

1. The sievert (Sv). The most important unusual unit in IPMB is the sievert. Both the sievert and the gray (Gy) are equal to a joule per kilogram. The gray is a physical unit measuring the energy deposited in tissue per unit mass, or the dose. The sievert is the gray multiplied by a dimensionless coefficient called the relative biological effectiveness and measures the effective dose. For x-rays, the sievert and gray are the same, but for alpha particles one gray can be many sieverts. An older unit for the gray is the rad (1 Gy = 100 rad) and an older unit for the sievert is the rem (1 Sv = 100 rem). The gray is named after English physicist Louis Gray, and the sievert after Swedish medical physicist Rolf Sievert.

Diffusion as a Random Walk

At the end of Chapter 4 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I model diffusion as a random walk.

The spreading solution to the one-dimensional diffusion equation that we verified can also be obtained by treating the motion of a molecule as a series of independent steps either to the right or to the left along the x axis.

Figure 4.24 in IPMB shows a simulation of a two-dimensional random walk that Russ added to the second edition.

Note how the particle wanders around one region of space and then takes a number of steps in the same direction to move someplace else. The particle trajectory is “thready.” It does not cover space uniformly. A uniform coverage would be very nonrandom. It is only when many particles are considered that a Gaussian distribution of particle concentration results.

I thought readers would profit from seeing the results of several simulations, so they won’t draw too many conclusions from one sample. Also, why let Russ have all the fun? So I wrote this MATLAB code, where “rand” is a random number generator with output between zero and one.

MATLAB code to perform a two-dimensional random walk.

Below I show nine different particle trajectories (plots of y versus x), for 40,000 steps (the same number Russ used in IPMB). The red dot indicates the starting location and the blue path shows the particle trajectory, which does look “thready.”

The particle trajectory for nine samples of a two-dimensional random walk,
each with 40,000 steps.

I also performed simulations for different numbers of steps, where r is the mean distance from the starting point to the end of the trajectory calculated by averaging over 10,000 samples. The red line is the result from continuum theory: distance equals the square root of the number of steps. The calculations agree with the theoretical prediction, but there is much scatter.

The average distance of the particle
from the starting point as a function of the number of steps,
for a two-dimensional random walk.

In IPMB we include an analogous result form Russ’s calculation, in which he averaged over only 328 samples, each with 10,000 steps. His results were within about a tenth of a percent of the theoretical prediction. Given the scatter in my simulations, I’m guessing Russ got lucky.

Sex-Linked Diseases

In Chapter 3 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I include a homework problem about color blindness.

Problem 6. Color blindness is a sex-linked defect. The defective gene is located in the X chromosome. Females carry an XX chromosome pair, while males have an XY pair. The trait is recessive, which means that the patient exhibits color blindness only if there is no normal X gene present. Let X_d be a defective gene. Then for a female, the possible gene combinations are

XX, XX_d, X_dX_d.

For a male, they are

XY, X_dY.

In a large population about 8% of the males are color-blind. What percentage of the females would you expect to be color-blind?

Textbook of Medical Physiology,
by Guyton and Hall.

In the Textbook of Medical Physiology (often cited in IPMB), Guyton and Hall write

Red-green color blindness is a genetic disorder that occurs almost exclusively in males. That is, genes in the female X chromosome code for the respective cones. Yet color blindness almost never occurs in females because at least one of the two X chromosomes almost always has a normal gene for each type of cone. Because the male has only one X chromosome, a missing gene can lead to color blindness.

Because the X chromosome in the male is always inherited from the mother, never from the father, color blindness is passed from mother to son, and the mother is said to be a color blindness carrier; this is true in about 8 per cent of all women.

Color blindness is not the only sex-linked defect. Many others exist, including hemophilia; an inability to clot blood. Those who suffer from hemophilia bleed profusely from minor cuts, and bruise easily. Guyton and Hall explain

Hemophilia is a bleeding disease that occurs almost exclusively in males. In 85 per cent of cases, it is caused by an abnormality or deficiency of Factor VIII; this type of hemophilia is called hemophilia A or classic hemophilia. About 1 of every 10,000 males in the United States has classic hemophilia. In the other 15 per cent of hemophilia patients, the bleeding tendency is caused by deficiency of Factor IX [hemophilia B]. Both of these factors are transmitted genetically by way of the female chromosome. Therefore, almost never will a woman have hemophilia because at least one of her two X chromosomes will have the appropriate genes. If one of her X chromosomes is deficient, she will be a hemophilia carrier, transmitting the disease to half of her male offspring and transmitting the carrier state to half of here female offspring.

Hemophilia B was common among the royal families of Europe in the 19th and 20th centuries. Queen Victoria of England was a carrier, and passed the mutation to royal houses in Spain, Germany and Russia. It may have played a role in triggering the Russian Revolution.