A Bit of Maths

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Trigonometric and exponential functions occur again and again in physics.  Let's look at them.  Here is a picture of some sinusoids.

SinesGreen.gif (6287 bytes)

Each curve is the derivative of the one  above it, that is, it shows the slope of the one above.  The first curve is a sine, the second is a cosine, and the third and fourth curves are the first two upside down.

So differentiating a sine gives this repeating pattern.  After four differentiations the curve is again a sine, the same as the first one.  Here is a picture of some exponentials.

ExpsGreen.gif (4542 bytes)

The curve sloping up (red) is exp(x) or ex, while that sloping down (blue) is exp(-x) or e-x.  In both cases the derivative or slope is the same as the original.  ex is the curve of compound interest, while e-x is curve of a radioactive decay, the fading of a fluorescent screen, or the dying away of the oscillation in an electrical tuned circuit, or the dying away of a piano tone with the damper off, or the dying away of the oscillation of a suspension bridge.

Sine and exponential functions are clearly different, yet the behaviour of their derivatives suggests a link.  If we slide a sine curve along, periodically it will fit over itself.  An exponential will not do that, but if we slide it any distance we like, and then multiply it by the right amount, it will look the same as before.  Sliding a distance means taking ex+k instead of ex, which means  ek x ex.  Therefore if we divide by ek we will be back where we started.  This means that    e-k x ex+k = ex.  Not what sine does, but not entirely different either.  Sliding most curves won't get you anywhere at all.  If we write these functions as infinite series we get -

ex = x0/0! + x1/1! + x2/2! + x3/3! +  +  +

sin(x) = x1/1! - x3/3! + x5/5! - x7/7! +  -  +

cos(x) = x0/0! - x2/2! + x4/4! - x6/6! +  -  +

These series are not entirely dissimilar except for the alternating signs in the trig functions.  The odd thing about the trig series is that although the curves are rigorously periodic, the series clearly aren't.  As x increases, none of the terms ever has the same value again.  Furthermore, if we try sin(100), we already have 1005 in the third term, 1007 in the fourth term, etc.  Considering that sin(x) is never bigger than 1, these are big numbers.  And what about sin(10000)?  There are going to be terms like 1000099/99! in the series.  These obviously almost cancel out in such a way that only the correct final result is left.  A small rounding error in the calculation would ruin the result.

Clearly the series is not the best way to calculate sin(x) for large x.  It is better to subtract a whole number of cycles, n x 2 x pi, from x, before using the series.

Can you prove, using the series, that sin(x + 2pi) = sin(x)

If we look at the series for sine and cosine we see that one has odd powers and one has even powers. This makes them antisymmetic and symmetric respectively about the y axis.  If we use y=ix in the exp series we get alternating signs, as in the other series, and in fact if we play our cards right, we get this -

eix = cos(x) + isin(x),

which suggests that exponentials and sines are quite closely related.

The next picture shows the values of the first twenty terms of the exponential series, for values of x up to 8.  The red line plots the maximum term for any x versus x + 0.5.

We see that the curve for a given x tends to be a maximum for the (x + 0.5)th term.  As x gets bigger, the curve of the terms tends towards a normal distribution.  From this we can derive an approximate formula for factorial n!  What is the formula?

Terms1Green.gif (13476 bytes)

The curves below show how the sine function is constructed from a series of terms in increasing powers of x.

SineTerms.gif (7903 bytes) SineTermsSymm.gif (8633 bytes)

The curves below show how the cosine function is constructed from a series of terms in increasing powers of x.

CosTerms.gif (7618 bytes)

The curves below show how the exponential function is constructed from a series of terms in increasing powers of x.

ExpTerms.gif (8496 bytes)

The curves below show how the same thing with the origin in the middle.

ExpTermsSymm.gif (8139 bytes)

The curves below show how the exp(-x) function is constructed from a series of terms in increasing powers of x.

ExpTermsNeg.gif (6096 bytes)

Then again, any repetitive signal can be constructed from a harmonic series of sinusoids . . . . .

You might wonder whether repetitive signals of any arbitrary shape, including sines, can be constructed from shapes other than sines.  Square waves for example.  The Rademacher functions and the Walsh functions are examples -

http://mathworld.wolfram.com/RademacherFunction.html

http://eyelid.ukonline.co.uk/nj71/walsh6.html

No doubt you can construct Rademacher functions and Walsh functions from sinusoids . . . .

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