IB Mathematics: Half-Range Expansions of Functions on a Finite Interval | IB数学：有限区间函数的半幅级数展开

📚 IB Mathematics: Half-Range Expansions of Functions on a Finite Interval | IB数学：有限区间函数的半幅级数展开

In IB Mathematics Analysis and Approaches HL, Fourier series are usually studied on symmetric intervals such as [-L, L]. However, many practical problems involve functions defined only on a finite interval [0, L]. A half-range expansion is a Fourier sine or cosine series that represents such a function on this half interval by artificially extending it to a full symmetric interval in an even or odd way.

在IB数学分析与方法HL中，傅里叶级数通常在对称区间如 [-L, L] 上研究。然而许多实际问题涉及仅在有限区间 [0, L] 上定义的函数。半幅级数展开通过将函数以偶方式或奇方式人工延拓到对称区间，从而用正弦级数或余弦级数在半个区间上表示原函数。

1. What Is a Half-Range Expansion? | 什么是半幅级数展开？

A half-range expansion expresses a function f(x) defined on 0 ≤ x ≤ L as an infinite series of sine terms only, or cosine terms only. The word “half-range” refers to the fact that the domain of the original function is only half of the full symmetric interval [-L, L] used in a standard Fourier series.

半幅级数展开将定义在 0 ≤ x ≤ L 上的函数 f(x) 表示为仅含正弦项或仅含余弦项的无穷级数。“半幅”一词是指原函数的定义域只是标准傅里叶级数所使用的对称区间 [-L, L] 的一半。

The two types are:

The half-range sine series: f(x) = Σ bₙ sin(nπx/L)
The half-range cosine series: f(x) = a₀/2 + Σ aₙ cos(nπx/L)

两种类型为：

半幅正弦级数：f(x) = Σ bₙ sin(nπx/L)
半幅余弦级数：f(x) = a₀/2 + Σ aₙ cos(nπx/L)

2. Periodic Extension and the Finite Interval | 周期延拓与有限区间

A Fourier series inherently represents a periodic function. Since a function on [0, L] is not periodic, we must choose a periodic extension after constructing a new function on the full interval [-L, L]. The extension determines the symmetry of the resulting series.

傅里叶级数本质上表示周期函数。由于 [0, L] 上的函数并非周期函数，我们必须先在完整区间 [-L, L] 上构造一个新函数，然后再进行周期延拓。延拓的方式决定了所得级数的对称性。

Given f on [0, L], we may define:

F(x) = f(x) for 0 ≤ x ≤ L, and F(-x) = f(x) for an even extension, or F(-x) = -f(x) for an odd extension. Then F is extended periodically with period 2L.

给定 [0, L] 上的 f，我们可以定义：

F(x) = f(x)（0 ≤ x ≤ L），且 F(-x) = f(x) 为偶延拓，或 F(-x) = -f(x) 为奇延拓。然后将 F 以 2L 为周期进行周期延拓。

3. Even Extension and the Cosine Series | 偶延拓与余弦级数

If F is an even function on [-L, L], all sine coefficients in its full Fourier series vanish. The remaining cosine series is called the half-range cosine series of f on [0, L].

如果 F 是 [-L, L] 上的偶函数，其完整傅里叶级数中所有正弦系数均为零。剩余的余弦级数称为 f 在 [0, L] 上的半幅余弦级数。

f(x) = a₀/2 + Σ aₙ cos(nπx/L), aₙ = (2/L)∫₀ᴸ f(x) cos(nπx/L) dx

The cosine series automatically satisfies zero derivative conditions at the endpoints, because the derivative of a cosine series is zero at x = 0 and x = L.

余弦级数自动满足端点零导数条件，因为余弦级数的导数在 x = 0 和 x = L 处为零。

4. Odd Extension and the Sine Series | 奇延拓与正弦级数

If F is an odd function on [-L, L], the cosine coefficients, including a₀, vanish. The resulting sine series is called the half-range sine series.

如果 F 是 [-L, L] 上的奇函数，所有余弦系数（包括 a₀）均为零。所得的正弦级数称为半幅正弦级数。

f(x) = Σ bₙ sin(nπx/L), bₙ = (2/L)∫₀ᴸ f(x) sin(nπx/L) dx

Because sin(nπx/L) = 0 at x = 0 and x = L, the sine series always satisfies zero value conditions at both endpoints.

因为 sin(nπx/L) 在 x = 0 和 x = L 处均为零，正弦级数总是满足两端点零值条件。

5. Deriving the Coefficients | 系数的推导

The formulas for aₙ and bₙ come from the orthogonality of sine and cosine functions over [0, L]. For example, ∫₀ᴸ cos(nπx/L) cos(mπx/L) dx = 0 when n ≠ m, and ∫₀ᴸ sin²(nπx/L) dx = L/2.

aₙ 和 bₙ 的公式来源于正弦、余弦函数在 [0, L] 上的正交性。例如，当 n ≠ m 时，∫₀ᴸ cos(nπx/L) cos(mπx/L) dx = 0，且 ∫₀ᴸ sin²(nπx/L) dx = L/2。

Type	Formula
Cosine coefficient aₙ, n ≥ 1	(2/L)∫₀ᴸ f(x) cos(nπx/L) dx
Constant term a₀	(2/L)∫₀ᴸ f(x) dx
Sine coefficient bₙ, n ≥ 1	(2/L)∫₀ᴸ f(x) sin(nπx/L) dx

Notice that the coefficient 2/L is used for every term, including a₀. A common mistake is to use 1/L instead.

注意每一项都使用系数 2/L，包括 a₀。常见错误是误用 1/L。

6. Worked Example: f(x) = x on [0, L] | 实例：f(x) = x 在 [0, L] 上

Let f(x) = x on the interval 0 ≤ x ≤ L. For the sine series, integrate by parts.

设 f(x) = x 在区间 0 ≤ x ≤ L 上。对于正弦级数，使用分部积分。

bₙ = (2/L)∫₀ᴸ x sin(nπx/L) dx = 2L(−1)ⁿ⁺¹ / (nπ)

Therefore the half-range sine series is:

x = Σₙ₌₁∞ [2L(−1)ⁿ⁺¹ / (nπ)] sin(nπx/L)

For the cosine series, a₀ = L and aₙ = 2L((−1)ⁿ − 1)/(n²π²). Thus the cosine series is:

x = L/2 − (4L/π²) Σₖ₌₀∞ [1/(2k+1)²] cos((2k+1)πx/L)

对余弦级数，a₀ = L，aₙ = 2L((−1)ⁿ − 1)/(n²π²)。因此余弦级数为：

x = L/2 − (4L/π²) Σₖ₌₀∞ [1/(2k+1)²] cos((2k+1)πx/L)

Check with L = π: x = π/2 − (4/π)(cos x + cos 3x/9 + cos 5x/25 + ⋯).

取 L = π 检验：x = π/2 − (4/π)(cos x + cos 3x/9 + cos 5x/25 + ⋯)。

7. Worked Example: f(x) = 1 on [0, L] | 实例：f(x) = 1 在 [0, L] 上

For f(x) = 1, the cosine series is trivial: a₀ = 2, aₙ = 0 for n ≥ 1, so f(x) = 1. The sine series is non-trivial.

对于 f(x) = 1，余弦级数是平凡的：a₀ = 2，n ≥ 1 时 aₙ = 0，所以 f(x) = 1。而正弦级数则非平凡。

bₙ = (2/L)∫₀ᴸ 1 · sin(nπx/L) dx = 2(1 − (−1)ⁿ) / (nπ)

Thus bₙ = 0 for even n, and bₙ = 4/(nπ) for odd n. The sine series is:

1 = (4/π) Σₖ₌₀∞ [1/(2k+1)] sin((2k+1)πx/L)

因此偶数 n 时 bₙ = 0，奇数 n 时 bₙ = 4/(nπ)。正弦级数为：

1 = (4/π) Σₖ₌₀∞ [1/(2k+1)] sin((2k+1)πx/L)

At x = L/2, this gives the Leibniz series: π/4 = 1 − 1/3 + 1/5 − 1/7 + ⋯.

在 x = L/2 处，可得莱布尼茨级数：π/4 = 1 − 1/3 + 1/5 − 1/7 + ⋯。

8. Convergence at Jump Discontinuities | 在跳跃间断点的收敛

At any point of continuity, the half-range series converges to f(x). At a point where the extended periodic function has a jump discontinuity, the series converges to the average of the left and right limits.

在连续点处，半幅级数收敛到 f(x)。在延拓后的周期函数具有跳跃间断的点处，级数收敛到左右极限的平均值。

Value = [f(x₀⁺) + f(x₀⁻)] / 2

For an odd extension of a function on [0, L], the endpoint x = 0 is often a jump discontinuity unless f(0) = 0. Similarly, x = L is a jump point unless f(L) = 0.

对于 [0, L] 上函数的奇延拓，除非 f(0) = 0，否则 x = 0 通常是跳跃间断点。同理，除非 f(L) = 0，x = L 也是跳跃点。

9. Gibbs Phenomenon | 吉布斯现象

When a half-range series has a jump discontinuity, the partial sums overshoot near the jump. This overshoot is approximately 9% of the jump size, and it does not diminish as the number of terms N increases; only the width of the overshoot region shrinks.

当半幅级数存在跳跃间断时，部分和在跳跃附近会出现过冲。过冲量约为跳跃量的 9%，并且随着项数 N 增加不会消失；只有过冲区域的宽度会变窄。

For example, the sine series of f(x) = 1 on [0, π] shows this effect near x = 0, where the odd extension jumps from 1 to −1.

例如，f(x) = 1 在 [0, π] 上的正弦级数在 x = 0 附近会出现该现象，因为奇延拓从 1 跳跃到 −1。

In IB exams, you are usually asked to identify the phenomenon or describe its qualitative behavior, not to calculate the exact overshoot value.

在IB考试中，通常只要求识别该现象或描述其定性行为，而不需要计算精确过冲值。

10. Choosing Between Sine and Cosine Expansions | 如何选择正弦或余弦展开

The choice depends on the boundary conditions suggested by the problem. A sine series is appropriate when f(0) = 0 and f(L) = 0, which corresponds to clamped or fixed endpoints in wave problems. A cosine series is appropriate when f′(0) = f′(L) = 0, which appears in insulated boundary conditions in heat conduction.

选择取决于问题所暗示的边界条件。当 f(0) = 0 且 f(L) = 0 时，适合使用正弦级数，对应于波动问题中的固定端。当 f′(0) = f′(L) = 0 时，适合使用余弦级数，出现在热传导中的绝热边界条件中。

Sine series: zero value at endpoints.
Cosine series: zero derivative at endpoints.

正弦级数：端点为函数零值。
余弦级数：端点为导数零值。

If no boundary condition is specified, both expansions are valid on the open interval (0, L), but they produce different values at the endpoints.

如果没有指定边界条件，两种展开在开区间 (0, L) 上都有效，但在端点处会给出不同的值。

11. Exam Tips and Common Pitfalls | 考试技巧与常见误区

Students frequently lose marks on half-range expansion questions for avoidable reasons.

学生在半幅级数展开题中常因可避免的原因丢分。

Always state whether you are using a sine or cosine expansion.
Use the coefficient 2/L, not 1/L.
For the cosine series, remember to compute a₀ separately using the same 2/L.
When n appears in the denominator, treat n = 0 separately.
At endpoints, use the midpoint convergence value rather than f itself.
Check whether the function is discontinuous in its odd or even extension.

务必说明使用的是正弦还是余弦展开。
使用系数 2/L，而不是 1/L。
对于余弦级数，记得单独计算 a₀，且同样使用 2/L。
当分母中出现 n 时，将 n = 0 单独处理。
在端点处，使用中点收敛值而不是 f 本身。
检查函数在偶延拓或奇延拓中是否是不连续的。

Also, remember that the same function can have two different half-range series; the sine and cosine series will generally differ, especially at the endpoints.

另外，同一函数可以有两个不同的半幅级数；正弦级数和余弦级数通常在端点处不同。

12. Summary | 总结

A half-range expansion is a powerful tool for representing functions on a finite interval. By choosing an even extension, we obtain a cosine series; by choosing an odd extension, we obtain a sine series. The coefficients are computed using integrals over [0, L] with the factor 2/L. Convergence is normal at continuity points, with the midpoint rule at jumps and the Gibbs phenomenon near jumps.

半幅级数展开是表示有限区间函数的强大工具。选择偶延拓得到余弦级数，选择奇延拓得到正弦级数。系数通过在 [0, L] 上取积分并乘以因子 2/L 计算。在连续点正常收敛，在跳跃点服从中点规则，跳跃附近出现吉布斯现象。

In IB examinations, mastering the coefficient formulas, the extension process, and the interpretation of convergence behaviour will enable you to solve these problems confidently and accurately.

在IB考试中，掌握系数公式、延拓过程以及收敛行为的解释，将帮助你自信而准确地解决这些问题。

Published by TutorHao | IB Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

IB Mathematics: Half-Range Expansions of Functions on a Finite Interval | IB数学：有限区间函数的半幅级数展开