Mathematica 2015

𝐶 ′ ( ) 𝐶( )

=  2 (2𝑖) 2 l[𝐶( )] = −4 2 2 2 𝐶( ) = 𝐶(0) −4 2 𝜋 2 𝑘 2

C(0) is the initial condition but now what is it? Let’s go back to the initial temperature distribution. ( ) = ( , 0) = ∑𝐶(0) 2𝜋𝑖𝑘𝑥 ∞ −∞ Surprisingly, we discover that C (0) is the coefficient of the Fourier expansion of f(x) . Then we know:

∫ ( ) 𝑖𝑘𝜋𝑥/𝐿 𝐿 −𝐿

1 2𝐿

𝐶(0) =



1 2

𝐿 =

Thus:

𝐶(0) = ∫ ( ) 2𝜋𝑖𝑘𝑥  1 2 −

1 2

Thus:

∫ ( ) 2𝜋𝑖𝑘𝑥  1 2 −

𝐶( ) =  −4 2 𝜋 2 𝑘 2

1 2

Finally, we write:

( , ) = ∑𝐶( ) 2𝜋𝑖𝑘𝑥 ∞ −∞

∫ ( ) 2𝜋𝑖𝑘𝑥  1 2 −

∞

( , ) = ∑ −4 2 𝜋 2 𝑘 2

 2𝜋𝑖𝑘𝑥

1 2

−∞

This formula looks quite long but still beautiful and impressive. It connects three variables together and reveals the relationship between temperature and time and position. We can know the temperature of the circular ring at any given time and position of the ring from this formula by simply plugging into the values of x and t that we want.



More about Fourier series:

The two examples above are very beautiful, but, still it is just a tip of an iceberg of Fourier series. A huge and fantastic world built upon Fourier series is still hidden behind it so that the unknown world motivates mathematicians to unravel

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