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Y ∂y Applying the mean value theorem and letting h tend to zero, we find that σ ∂u ∂2u ∂2u = + 2; κ ∂t ∂x2 ∂y this is called the time-dependent heat equation, often abbreviated to the heat equation. 2 Steady-state heat equation in the disc After a long period of time, there is no more heat exchange, so that the system reaches thermal equilibrium and ∂u/∂t = 0. In this case, the time-dependent heat equation reduces to the steady-state heat equation (10) ∂2u ∂2u + 2 = 0. ∂x2 ∂y The operator ∂ 2 /∂x2 + ∂ 2 /∂y 2 is of such importance in mathematics and physics that it is often abbreviated as and given a name: the Laplace operator or Laplacian.

Here, it is important to pin down where our function is originally defined. If f is an integrable function given on an interval [a, b] of length L (that is, b − a = L), then the nth Fourier coefficient of f is defined by 1 fˆ(n) = L b f (x)e−2πinx/L dx, n ∈ Z. a The Fourier series of f is given formally3 by ∞ fˆ(n)e2πinx/L . n=−∞ We shall sometimes write an for the Fourier coefficients of f , and use the notation ∞ an e2πinx/L f (x) ∼ n=−∞ to indicate that the series on the right-hand side is the Fourier series of f.

Ibookroot 24 October 20, 2007 Chapter 1. THE GENESIS OF FOURIER ANALYSIS 3. A sequence of complex numbers {wn }∞ n=1 is said to converge if there exists w ∈ C such that lim |wn − w| = 0, n→∞ and we say that w is a limit of the sequence. (a) Show that a converging sequence of complex numbers has a unique limit. The sequence {wn }∞ n=1 is said to be a Cauchy sequence if for every > 0 there exists a positive integer N such that |wn − wm | < whenever n, m > N . (b) Prove that a sequence of complex numbers converges if and only if it is a Cauchy sequence.