By Aristoxenus, Henry Stewart Macran
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Additional info for Aristoxenoy Armonika stoicheia: The harmonics of Aristoxenus
6 ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ 4. Find the Fourier series of the continuous periodic triangular wave function defined by π −θ 0≤θ ≤π f (θ) = 2 3π θ − 2 π ≤ θ ≤ 2π and f (θ + 2π) = f (θ). ✁ ✁❆ ❆ ❆ ❆ ❆ ❆✁ ✁ ✁ ✁ ✁ ✁❆ ❆ ❆ ❆ ❆ ✁ ❆✁ ✁ ✁ ✁❆ ✁ ❆ ❆ ❆ ❆ ✁ ❆✁ ✁ ✁ ✁❆ ✁ ❆ ❆ ❆ ❆ ❆ 5. 7) are bounded. (b) If f (θ) is a differentiable periodic function with period 2π, find the relationship between the Fourier coefficients am (f ), bm (f ) for f (θ) and the Fourier coefficients am (f ′ ), bm (f ′ ) for the derivative f ′ (θ).
4. Let α be a real number with |α| < 1. Use complex exponentials, as in Appendix n C, to find the sum of the Fourier series ∞ n=1 α sin nθ. 3. Even and odd functions A function f (θ) is said to be even if f (−θ) = f (θ), and it is said to be odd if f (−θ) = −f (θ). For example, cos θ is even, while sin θ is odd. Of course, most functions are neither even nor odd. If a function happens to be both even and odd, then it is zero, because we have f (θ) = f (−θ) = −f (θ). , 12 (f (θ) + f (−θ)). Similarly, 21 (f (θ) − f (−θ)) is an odd function.
If not, explain why not. (i) sin θ + sin 45 θ. √ (ii) sin θ + sin 2 θ. (iii) sin2 θ. (iv) sin(θ2 ). (v) sin θ + sin(θ + π3 ). 3. Draw graphs of the functions sin(220πt) + 21 sin(440πt) and sin(220πt) + 1 2 cos(440πt). Explain why these sound the same, even though the graphs look quite different. 4. Let α be a real number with |α| < 1. Use complex exponentials, as in Appendix n C, to find the sum of the Fourier series ∞ n=1 α sin nθ. 3. Even and odd functions A function f (θ) is said to be even if f (−θ) = f (θ), and it is said to be odd if f (−θ) = −f (θ).