📚 Functions Defined on a Finite Range: Half-Range Series | 定义在有限区间上的函数：半幅级数
In many physical and engineering problems, a function is known only over a finite interval, such as [0, L]. Fourier series traditionally represent periodic functions, but by cleverly extending the given function beyond its original domain, we can construct a periodic expansion that matches the function on [0, L] and involves only sine terms or only cosine terms. These are called half-range series, and they are indispensable tools in solving partial differential equations with boundary conditions.
在许多物理和工程问题中，函数仅在有限区间（如 [0, L]）上是已知的。傅里叶级数通常用于表示周期函数，但通过巧妙地延拓原函数到区间之外，我们可以构造一个周期展开式，使其在 [0, L] 上与已知函数一致，并且只包含正弦项或只包含余弦项。这些级数称为半幅级数，它们是求解带有边界条件的偏微分方程时不可或缺的工具。
1. The Need for Half-Range Expansions | 半幅展开的必要性
A standard Fourier series of a function f(x) with period 2L contains both sine and cosine terms. However, if we only have data on [0, L], we are free to choose how the function behaves on [-L, 0] as long as the resulting periodic extension converges to the correct values on [0, L]. Half-range expansions exploit this freedom to simplify the representation, often aligning with physical symmetries such as a string fixed at one end or a rod insulated at an extremity.
周期为 2L 的函数 f(x) 的标准傅里叶级数同时包含正弦和余弦项。然而，如果我们只有 [0, L] 上的信息，我们就可以自由选择函数在 [-L, 0] 上的行为，只要最终得到的周期延拓在 [0, L] 上收敛到正确的值。半幅展开正是利用这种自由来简化表达式，这往往与物理对称性（如一端固定的弦或一端绝热的杆）相吻合。
2. Even and Odd Extensions | 偶延拓与奇延拓
To obtain a half-range cosine series, we create an even extension of f(x) by reflecting the graph about the vertical axis. This yields an even function on [-L, L] that can be extended periodically with period 2L. For a half-range sine series, we construct an odd extension by reflecting the graph through the origin, producing an odd function on [-L, L]. The choice of extension dictates which Fourier coefficients survive.
为了获得半幅余弦级数，我们通过将图像关于纵轴反射来构造 f(x) 的偶延拓，从而在 [-L, L] 上得到一个偶函数，并可将其周期延拓为周期 2L 的函数。对于半幅正弦级数，我们通过将图像关于原点反射来构造奇延拓，生成 [-L, L] 上的奇函数。延拓方式决定了哪些傅里叶系数会保留下来。
3. Half-Range Cosine Series | 半幅余弦级数
If f(x) is given on [0, L] and we extend it as an even function of period 2L, the resulting Fourier series contains only cosine terms (and a constant term). The half-range cosine series of f(x) on [0, L] is:
如果 f(x) 给定在 [0, L] 上，并且我们将其延拓为周期 2L 的偶函数，则得到的傅里叶级数只包含余弦项（以及常数项）。f(x) 在 [0, L] 上的半幅余弦级数为：
f(x) = a₀/2 + ∑n=1∞ aₙ cos(nπx / L)
where the coefficients aₙ are determined only by the values of f on [0, L]. The even extension ensures that all sine coefficients vanish identically.
其中系数 aₙ 仅由 f 在 [0, L] 上的值确定。偶延拓保证了所有正弦系数恒为零。
4. Half-Range Sine Series | 半幅正弦级数
By constructing an odd periodic extension of f(x) with period 2L, the Fourier series reduces to a pure sine series. The half-range sine series on [0, L] takes the form:
通过构造周期为 2L 的奇周期延拓，傅里叶级数简化为纯正弦级数。在 [0, L] 上的半幅正弦级数形式为：
f(x) = ∑n=1∞ bₙ sin(nπx / L)
presentation is particularly useful when boundary conditions require f(0) = 0 or f(L) = 0.
请注意，常数项 a₀/2 消失了，因为奇函数在对称区间上的平均值为零。这种表示在边界条件要求 f(0) = 0 或 f(L) = 0 时特别有用。
5. Computing Cosine Coefficients | 半幅余弦系数计算
The Fourier cosine coefficients are obtained by integrating the even extension over a full period. Because the integrand is even, the integral from -L to L simplifies to twice the integral from 0 to L. This gives the well-known formula:
傅里叶余弦系数是通过对偶延拓在完整周期上积分得到的。由于被积函数是偶函数，从 -L 到 L 的积分简化为从 0 到 L 积分的两倍。由此得到著名的公式：
a₀ = (2/L) ∫0L f(x) dx
aₙ = (2/L) ∫0L f(x) cos(nπx / L) dx, n = 1, 2, 3, ...
These coefficients guarantee that the cosine series converges to f(x) at points of continuity when the even extension is piecewise smooth.
这些系数保证了当偶延拓分段光滑时，余弦级数在连续点处收敛于 f(x)。
6. Computing Sine Coefficients | 半幅正弦系数计算
For the half-range sine series, the odd extension leads to an odd integrand over [-L, L]. The integral of an odd function over a symmetric interval is twice the integral from 0 to L. Hence the sine coefficients are:
对于半幅正弦级数，奇延拓导致在 [-L, L] 上的被积函数为奇函数。奇函数在对称区间上的积分是从 0 到 L 积分的两倍。因此正弦系数为：
bₙ = (2/L) ∫0L f(x) sin(nπx / L) dx, n = 1, 2, 3, ...
Since bₙ depends only on f(x) on [0, L], the sine expansion is entirely determined by the given data. At the endpoints, the sine series converges to zero irrespective of the actual values of f(0+) and f(L-), which is a characteristic feature.
由于 bₙ 仅取决于 f(x) 在 [0, L] 上的值，正弦展开完全由给定数据决定。在端点处，无论 f(0⁺) 和 f(L⁻) 的实际值是多少，正弦级数都收敛到零，这是一个显著特征。
7. Worked Example: Cosine Series for f(x) = x on [0, π] | 示例：f(x) = x 在 [0, π] 上的余弦级数
(2/π) ∫₀^π x dx = (2/π)[x²/2]₀^π = π. For n ≥ 1, we compute aₙ using integration by parts.
考虑定义在 [0, π] 上的 f(x) = x。为求其半幅余弦级数，令 L = π。系数 a₀ 为 (2/π) ∫₀^π x dx = (2/π)[x²/2]₀^π = π。对于 n ≥ 1，我们使用分部积分计算 aₙ。
aₙ = (2/π) ∫0π x cos(nx) dx = (2/π) [ (x sin(nx))/n + (cos(nx))/n² ]0π
= 0; for odd n = 2k-1, aₙ = -4/(π (2k-1)²). Thus the cosine series is f(x) = π/2 - (4/π) ∑k=1^∞ cos((2k-1)x)/(2k-1)².
代入上下限，sin(nπ) = 0，cos(nπ) = (-1)ⁿ，而在 x = 0 处 cos(0)=1。得到 aₙ = (2/π)[((-1)ⁿ - 1)/n²] = (2/π) * ((-1)ⁿ - 1)/n²。对于偶数 n，aₙ = 0；对于奇数 n = 2k-1，aₙ = -4/(π (2k-1)²)。因此余弦级数为 f(x) = π/2 - (4/π) ∑k=1^∞ cos((2k-1)x)/(2k-1)²。
8. Worked Example: Sine Series for f(x) = x on [0, π] | 示例：f(x) = x 在 [0, π] 上的正弦级数
he sine coefficients bₙ are given by (2/π) ∫₀^π x sin(nx) dx. Integration by parts yields:
使用同一个在 [0, π] 上的函数，现在求其半幅正弦级数。正弦系数 bₙ 由 (2/π) ∫₀^π x sin(nx) dx 给出，分部积分得到：
bₙ = (2/π) [ - (x cos(nx))/n + (sin(nx))/n² ]0π
1)ⁿ / n ] = -2(-1)ⁿ / n = 2(-1)ⁿ⁺¹ / n. The half-range sine series is therefore f(x) = 2 ∑n=1 series converges to 0 while f(π) = π, showing the convergence to the midpoint of the jump.
在 x = π 处，cos(nπ) = (-1)ⁿ，sin(nπ)=0；在 x = 0 处，cos(0)=1 且该项消失。得到 bₙ = (2/π)[ -π(-1)ⁿ / n ] = -2(-1)ⁿ / n = 2(-1)ⁿ⁺¹ / n。因此半幅正弦级数为 f(x) = 2 ∑n=1^∞ ((-1)ⁿ⁺¹ / n) sin(nx)。在 x = π 处，级数收敛到 0，而 f(π) = π，显示了在跳跃间断点处级数收敛到跃度中点。
9. Convergence and Gibbs Phenomenon | 收敛性与吉布斯现象
Half-range series inherit the convergence properties of classical Fourier series. At points where the periodic extension is continuous, the series converges to f(x). At jump discontinuities—such as at the endpoints for a sine extension when f(0) or f(L) is not zero—the series converges to the average of the left- and right-hand limits. Moreover, partial sums exhibit Gibbs phenomenon: near a jump, overshoots of about 9% of the jump size persist no matter how many terms are taken.
半幅级数继承了经典傅里叶级数的收敛性质。在周期延拓连续的点，级数收敛于 f(x)。在跳跃间断点处——例如当正弦延拓在 f(0) 或 f(L) 非零时的端点——级数收敛到左右极限的平均值。此外，部分和会表现出吉布斯现象：在跳跃点附近，无论取多少项，总会出现约 9% 跳跃幅度的过冲。
cated series overshoots the value 0 near the discontinuity, an effect that has profound implications in signal processing and control theory.
这在 f(x)=x 在 x=π 处的正弦级数中尤为明显。截断级数在间断点附近会超过 0 值，这一现象在信号处理和控制理论中具有深远影响。
10. Choosing Between Sine and Cosine Series | 正弦与余弦级数的选择
tion problem with insulated ends, the cosine series ensures zero derivative at the boundaries. When no physical constraint is present, both series are mathematically valid, but one may converge faster depending on the smoothness of the extension.
在实际应用中，边界条件往往决定了半幅展开的选择。对于两端固定的振动弦，正弦级数自然满足在 x = 0 和 x = L 处位移为零的条件。对于两端绝热的热传导问题，余弦级数可保证边界处的导数为零。在没有物理约束时，两种级数在数学上都是有效的，但视延拓后的光滑性不同，某一级数可能收敛得更快。
11. Connection to Partial Differential Equations | 与偏微分方程的联系
Half-range series are a cornerstone in solving linear PDEs by separation of variables. For example, the one-dimensional wave equation utt = c² uxx on [0, L] with fixed ends leads to a half-range sine series in x. The heat equation with Neumann boundary conditions leads to a half-range cosine series. The coefficients found from initial conditions are exactly the Fourier coefficients of the half-range expansion of the initial profile.
半幅级数是使用分离变量法求解线性偏微分方程的基石。例如，在 [0, L] 上带有固定端点的波动方程 utt = c² uxx 会得到关于 x 的半幅正弦级数。带有诺伊曼边界条件的热方程则导出半幅余弦级数。由初始条件求出的系数恰好就是初始分布半幅展开的傅里叶系数。
12. Summary and Key Takeaways | 要点总结
Half-range series provide a powerful method to represent a function defined on a finite interval purely in terms of sines or cosines. The key steps are: (i) choose an even or odd extension, (ii) compute the appropriate Fourier coefficient integrals over [0, L], and (iii) write the series. Understanding the implications of the chosen extension on convergence and boundary behavior is essential for successful application in engineering mathematics and beyond.
半幅级数为仅定义在有限区间上的函数提供了一种纯粹用正弦或余弦表示的强有力方法。关键步骤是：(i) 选择偶延拓或奇延拓，(ii) 在 [0, L] 上计算相应的傅里叶系数积分，(iii) 写出级数。理解所选延拓对收敛性和边界行为的影响，对于在工程数学及其他领域中成功应用半幅级数至关重要。
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