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        "intro": "<p>In this Section we show how a Fourier series can be expressed more concisely if we introduce the complex number \\(i\\) where \\(i^2 = -1\\). By utilising the Euler relation: \\[ e^{i\\theta} \\equiv \\cos \\theta + i \\sin \\theta \\]<\/p>\n<p>we can replace the trigonometric functions by complex exponential functions. By also combining the Fourier coefficients \\(a_n\\) and \\(b_n\\) into a complex coefficient \\(c_n\\) through \\[ c_n = \\dfrac{1}{2} \\left( a_n - i b_n \\right) \\]<\/p>\n<p>we find that, for a given periodic signal, both sets of constants can be found in one operation.<\/p>\n<p>We also obtain Parseval's theorem which has important applications in electrical engineering.<\/p>\n<p>The complex formulation of a Fourier series is an important precursor of the Fourier transform which attempts to Fourier analyse non-periodic functions.<\/p>",
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