A Bit of Maths Back to Home page back to Oscillation back to Nature's Maths Trigonometric and exponential functions occur again and again in physics. Let's look at them. Here is a picture of some sinusoids. 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. The curve sloping up (red)
is exp(x) or e 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 e e sin(x) = x cos(x) = x 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 100 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 - e 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? The curves below show how the sine function is constructed from a series of terms in increasing powers of x. The curves below show how the cosine function is constructed from a series of terms in increasing powers of x. The curves below show how the exponential function is constructed from a series of terms in increasing powers of x. The curves below show how the same thing with the origin in the middle. The curves below show how the exp(-x) function is constructed from a series of terms in increasing powers of x. 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 . . . . Back to Home page back to Oscillation back to Nature's Maths |