Sine and Cosine Integrals and the Delta Function

Trigger warning: This post is for mature audiences only; it may contain Fourier transforms and Dirac delta functions. 

In Chapter 11 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I examine some properties of Fourier transforms. In particular, we consider three integrals of sines and cosines. After some analysis, we conclude that these integrals are related to the Dirac delta function, δ(ω – ω’), equal to infinity at ω = ω’ and zero everywhere else (it’s a strange function consisting of one really tall, thin spike).

Are these equations correct? I now believe that they’re almost right, but not entirely. I propose that instead they should be 

You’re probably thinking “what a pity, the second three equations looks more complicated than the first three.” I agree. But let me explain why I think they’re better. Hang on, it’s a long story.

Let’s go back to our definition of the Fourier transform in Eq. 11.57 of IPMB

The first thing to note is that y(t) consists of two parts. The first depends on cos(ωt), which is an even function, meaning cos(–ωt) = cos(ωt). There’s an integral over ω, implying that many different frequencies contribute to y(t), weighted by the function C(ω). But one thing we know for sure is that when you add up the contributions from all these many frequencies, the result must be an even function (the sum of even functions is an even function). The second part depends on sin(ωt), which is an odd function, sin(–ωt) = – sin(ωt). Again, when you add up all the contributions from these many frequencies weighted by S(ω), you must get an odd function. So, we can say that we’re writing y(t) as the sum of an even part, y_even(t), and an odd part, y_odd(t). In that case, we can rewrite our Fourier transform expressions as

We should be able to take our expression for y_even(t), put our expression for C(ω) into it, and then—if all works as it should—get back y_even(t). Let’s try it and see if it works. To start I’ll just rewrite the first of the four equations listed above

Now for C(ω) I’ll use the third of the four equations listed above. In that expression, there is an integral over t, but t is a dummy variable (it’s an “internal” variable; after you do the integral, the result does not depend on t), so to keep things from getting confusing we’ll call the dummy variable by another name, t'

Next switch the order of the integrals, so the integral over t' is on the outside and the integral over ω is on the inside

Ha! There, inside the bracket, is one of those integrals were’re talking about. Okay, the variables ω and t are swapped, but otherwise it’s the same. So, let’s put in our new expression for the integral

The 2π’s cancel, and a factor of one half comes out. An integral containing a delta function just picks out the value where the argument of the delta function is zero. We get

But, we know that y_even(t) is an even function, meaning y_even(–t) equals y_even(t). So finally

It works! We go “around the loop” and get back our original function.

You could perform another calculation just like this one but for y_odd(t). Stop reading and do it, to convince yourself that again you get back to where you started from, y_odd(t) = y_odd(t).

Now, you folks who are really on the ball might realize that if you had used the old delta function relationships given in IPMB (the first three equations in this post), they would also work. (Again, try it and see.) So why use my fancy new formulas? Well, if you have an integral that adds up a bunch of cos(ωt), you know you’re gonna get an even function. There’s no way it can be equal to δ(ω – ω’), because that function is neither even nor odd. So, it just doesn’t make sense to say that summing up a bunch of even functions gives you something that isn’t even. In my new formula, that sum of two delta functions is an even function. The same argument holds for the integral with sin(ωt), which must be odd.

Finally (and this is what got me started down this rabbit hole), you often see the delta function written as

Jackson even gives this equation, so it MUST be correct. (For those of who aren’t physicists, John David Jackson wrote the highly regarded graduate textbook Classical Electrodynamics, known by physics graduate students simply as “Jackson.”)

In Jackson’s equation, i is the square root of minus one. So, this representation of the delta function uses complex numbers. You won’t see it in IPMB because Russ and I avoid complex numbers (I hate them).

Let’s use the Euler formula e^iθ = cosθ + i sinθ to change the integral in Jackson’s delta function expression to

Now use a couple trig identities, cos(A–B) = cosA cosB + sinA sinB and sin(A–B) = sinA cosB –cosA sinB, to get

This is really four integrals,

Then, using the relations between these integrals and the delta function given in IPMB (the first three equations at the top of this post), you get that the sum of these integrals is equal to

which is obviously wrong; we started with 2πδ and ended up with 4πδ. Even worse, do the same calculation for δ(ω + ω') with a plus instead of a minus in front of the ω'. I’ll leave it to you to work out the details, but you’ll get zero! Again, nonsense. However, if you use the integral relations I propose above (second set of three integrals at the top of this blog), everything works just fine (try it).

Gene Surdutovich, my new coauthor for the sixth edition of IPMB, and I are still deciding how to discuss this issue in the new edition (which we are hard at work on but I doubt will be out within the next year). I don’t want to get bogged down in mathematical minutia that isn’t essential to our book’s goals, but I want our discussion to be correct. Once the sixth edition is published, you can see how we handle it.

I haven’t seen my new delta function/Fourier integral relationships in any other textbook or math handbook. This makes me nervous. Are they correct? Moreover, Intermediate Physics for Medicine and Biology does not typically contain new mathematical results. Maybe I haven’t looked hard enough to see if someone else published these equations (if you’ve seen them before, let me know where…please!). Maybe I’ll find these results in Morse and Feshbach (another one of those textbooks known to all physics graduate students) or some other mathematical tome. I need to make a trip to the Oakland University library to look through their book collection, but right now its too cold and snowy (we got about four to five inches of the white stuff in the last 48 hours).