📚 Obtaining one Fourier series from another | 由一个傅里叶级数推导另一个傅里叶级数
Once you have established the Fourier series expansion of a particular function, you can often derive the series for many other functions without having to evaluate the Fourier coefficients from scratch. By applying operations such as linear combination, change of variable, differentiation, integration, and exploiting symmetry, one known Fourier series can be transformed into another. This technique saves time and provides deeper insight into the structure of Fourier expansions.
一旦你确定了某个函数的傅里叶级数展开，往往不需要从头计算傅里叶系数，就能推导出许多其他函数的级数。通过线性组合、变量替换、微分、积分以及利用对称性等操作，一个已知的傅里叶级数可以变换成另一个。这种方法既节省时间，又能加深对傅里叶展开结构的理解。
1. Review of standard Fourier series forms | 标准傅里叶级数形式回顾
A real-valued periodic function f(x) with period 2π can be represented as f(x) ∼ n nx dx. For a period 2L, the series uses cos(nπx/L) and sin(nπx/L). In complex form, f(x) ∼ Σₙ₌₋∞∞ cₙ einx with cₙ = (1/2π) ∫₋ππ f(x) e−inx dx.
一个周期为 2π 的实值函数 f(x) 可表示为 f(x) ∼ a₀/2 + Σₙ₌₁∞ (aₙ cos nx + bₙ sin nx)，其中 aₙ = (1/π) ∫₋ππ f(x) cos nx dx，bₙ = (1/π) ∫₋ππ f(x) sin nx dx。若周期为 2L，则采用 cos(nπx/L) 和 sin(nπx/L)。复指数形式为 f(x) ∼ Σₙ₌₋∞∞ cₙ einx，cₙ = (1/2π) ∫₋ππ f(x) e−inx dx。
2. Linear combinations of known series | 通过已知级数的线性组合获得新级数
y for bₙ. No new integration is needed. This is especially useful when a target function can be decomposed into elementary functions with known expansions.
如果已知 f(x) 和 g(x) 的傅里叶级数，那么 h(x) = αf(x) + βg(x) 的级数就是各个级数的相同线性组合：aₙ(h) = α aₙ(f) + β aₙ(g)，bₙ 同理。无需重新积分。当目标函数可以分解为具有已知展开的基本函数时，这一方法尤其好用。
x + C simply adds a constant C to the a₀/2 term, leaving all other coefficients unchanged.
例如，设 f(x) = x (−π < x < π) 的级数为 2 Σₙ₌₁∞ ((−1)ⁿ⁺¹/n) sin nx，则 h(x) = x + C 只需将常数 C 加到 a₀/2 项上，其余系数不变。
3. Change of variable: scaling the argument | 变量替换：自变量的缩放
Given the 2π-periodic series for f(x), the series for f(kx) is obtained by replacing x with kx in the expansion. If f(x) ∼ Σₙ cₙ einx, then f(kx) ∼ Σₙ cₙ einkx, which is a Fourier series on a new period 2π/|k|. In terms of standard period 2π, this corresponds to a series with non-integer harmonics unless k is an integer.
已知 f(x) 的 2π 周期级数，f(kx) 的级数只需在展开式中将 x 替换为 kx。若 f(x) ∼ Σₙ cₙ einx，则 f(kx) ∼ Σₙ cₙ einkx，这是一个以 2π/|k| 为新周期的傅里叶级数。若用标准周期 2π 表示，除非 k 为整数，否则谐波频率将不是整数倍。
tiples of k. For instance, the square wave expansion sin x + (1/3) sin 3x + (1/5) sin 5x + … yields the series for sin 2x + (1/3) sin 6x + … by simply setting k=2.
当 k 为整数时，f(kx) 继承相同的系数，但只出现在 k 的倍数处。例如，方波展开 sin x + (1/3) sin 3x + (1/5) sin 5x + …，取 k=2 即可得到 sin 2x + (1/3) sin 6x + … 的级数。
4. Translation and reflection of the variable | 自变量的平移与反射
Shifting the argument x → x − x₀ alters the phase of each harmonic. If f(x) ∼ a₀/2 + Σ (aₙ cos nx + bₙ sin nx), then f(x − x₀) ∼ cos nx₀) sin nx ]. This reveals how the Fourier coefficients transform under translation.
对自变量进行平移 x → x − x₀ 会改变每个谐波的相位。若 f(x) ∼ a₀/2 + Σ (aₙ cos nx + bₙ sin nx)，则 f(x − x₀) ∼ a₀/2 + Σ [ (aₙ cos nx₀ − bₙ sin nx₀) cos nx + (aₙ sin nx₀ + bₙ cos nx₀) sin nx ]，这展示了平移下傅里叶系数的变化规律。
Reflection f(−x) simply changes the sign of sine coefficients and keeps cosine coefficients intact (if f is defined across symmetric intervals). Therefore, if you know the series of f(x), you can write down the series of f(−x) instantly without any integration.
反射 f(−x) 仅改变正弦系数的符号，余弦系数保持不变（前提是 f 定义在对称区间上）。因此，若已知 f(x) 的级数，便可以立刻写出 f(−x) 的级数，无需任何积分。
5. Term-by-term differentiation | 逐项微分
Under certain smoothness conditions, the Fourier series of f'(x) can be obtained by differentiating the series of f(x) term by term. If f(x) ∼ a₀/2 + Σ (aₙ cos nx + bₙ sin nx) and f is continuous and piecewise smooth, then f'(x) ∼ Σ (−n aₙ sin nx + n bₙ cos nx). This gives a new series with coefficients multiplied by n.
在一定光滑性条件下，f'(x) 的傅里叶级数可以通过对 f(x) 的级数逐项微分得到。若 f(x) ∼ a₀/2 + Σ (aₙ cos nx + bₙ sin nx) 且 f 连续且分段光滑，则 f'(x) ∼ Σ (−n aₙ sin nx + n bₙ cos nx)，新级数的系数乘以 n。
This is extremely powerful: from the known series of a triangle wave, which is continuous, you can differentiate to obtain the series of a square wave, up to a constant factor. The constant term a₀/2 disappears under differentiation.
该方法极为有效：已知连续三角波的级数后，可通过微分（差一个常数因子）得到方波的级数。微分后常数项 a₀/2 消失。
6. Term-by-term integration | 逐项积分
elds a convergent series, even if the original series is not uniformly convergent. If f(x) ∼ a₀/2 + Σ (aₙ cos nx + bₙ sin nx), then ∫₀ˣ f(t) dt ∼ nx)/n − Σ (bₙ cos nx)/n + C, where the constant C is determined by the initial condition.
相反地，对傅里叶级数逐项积分总能得到一个收敛的级数，即便原级数并不一致收敛。若 f(x) ∼ a₀/2 + Σ (aₙ cos nx + bₙ sin nx)，则 ∫₀ˣ f(t) dt ∼ a₀x/2 + Σ (aₙ sin nx)/n − Σ (bₙ cos nx)/n + C，常数 C 由初始条件确定。
, one often subtracts this secular term. This technique can transform a square wave series into a triangular wave series.
积分会引入线性项 a₀x/2，它并非周期项，除非 a₀ = 0。为得到纯粹的傅里叶级数，通常需减去这个长期项。此技巧可将方波级数转化为三角波级数。
7. Exploiting even and odd symmetry | 利用奇偶对称性
An even function has only cosine terms, and an odd function has only sine terms. If you know the full series of a function, you can extract its even part [f(x)+f(−x)]/2 to obtain a pure cosine series, and its odd part [f(x)−f(−x)]/2 for a pure sine series. This effectively provides two different Fourier expansions from one known series.
偶函数仅含余弦项，奇函数仅含正弦项。若已知某个函数的完整级数，可取其偶部 [f(x)+f(−x)]/2 得到纯余弦级数，取其奇部 [f(x)−f(−x)]/2 得到纯正弦级数。这样就能从一个已知级数有效得到两种不同的傅里叶展开。
Moreover, by shifting a known series to make it symmetric about zero, you can often simplify the coefficient pattern and relate it to the series of a similar function on a half-range expansion.
此外，通过平移已知级数使其关于原点对称，往往能简化系数模式，并将其与类似函数在半区间展开的级数联系起来。
8. Adapting a known series to a new period | 将已知级数适配到新的周期
] by a simple scaling substitution. For a period 2L, the series becomes a₀/2 + Σ [ aₙ cos(nπx/L) + bₙ sin(nπx/L) ] with the same aₙ, bₙ. This allows one standard expansion to serve many different scales.
如果已有 f(x) 在 [−π, π] 上的傅里叶级数，则可通过缩放替换 g(x) = f(πx/L) 得到区间 [−L, L] 上的级数。对于周期 2L，级数变为 a₀/2 + Σ [ aₙ cos(nπx/L) + bₙ sin(nπx/L) ]，aₙ、bₙ 保持不变。一个标准展开因此适用于多种尺度。
For example, knowing the expansion of |x| on [−π, π], you can immediately write the expansion of |x| on [−L, L] by scaling the argument. No recalculation of coefficients is required.
例如，已知 |x| 在 [−π, π] 上的展开，通过缩放自变量即可直接写出 |x| 在 [−L, L] 上的展开，无需重新计算系数。
9. Using Parseval's identity to derive series sums | 利用 Parseval 恒等式推导级数和
Parseval' res of its Fourier coefficients. Once you have a Fourier series, you can evaluate 1/π ∫₋ππ |f(x)| y or by comparing with a known integral, you can deduce the sum of a new numerical series.
Parseval 恒等式将函数的均方值与其傅里叶系数平方和联系起来。一旦有了一个傅里叶级数，便可计算 1/π ∫₋ππ |f(x)|² dx = a₀²/2 + Σₙ₌₁∞ (aₙ² + bₙ²)。通过适当选择 x 或与已知积分比较，可以推导出一个新的数项级数的和。
This method is often used to obtain sums like Σ 1/n², Σ 1/n⁴, etc., from elementary Fourier series. It provides an elegant bridge between trigonometric series and famous constants.
该方法常用于从基本傅里叶级数得到如 Σ 1/n²、Σ 1/n⁴ 等和式，在三角级数与著名常数之间搭建起一座优美的桥梁。
10. Complex exponential form manipulations | 复指数形式的操作
Working with the complex form f(x) ∼ Σ cₙ einx makes many operations cleaner. Conjugation gives the series for the complex conjugate function: f̄(x) ∼ Σ c̄₋ₙ einx. Shifting the index yields the series for f(x) eimx as Σ cₙ₋ₘ einx. These algebraic manipulations avoid trigonometric identities.
使用复指数形式 f(x) ∼ Σ cₙ einx 可使许多操作更加简洁。取共轭即得共轭函数的级数：f̄(x) ∼ Σ c̄₋ₙ einx。移动下标可得 f(x) eimx 的级数为 Σ cₙ₋ₘ einx。这些代数操作避开了三角恒等式的复杂性。
Multiplying two Fourier series in complex form corresponds to the convolution of their coefficient sequences: if f ∼ Σ cₙ einx and g ∼ Σ dₙ einx, then f·g ∼ Σ (c ∗ d)ₙ einx where (c ∗ d)ₙ = Σₖ cₖ dₙ₋ₖ. This provides a systematic way to derive the series of a product.
两个复指数形式傅里叶级数的乘积对应于其系数序列的卷积：若 f ∼ Σ cₙ einx，g ∼ Σ dₙ einx，则 f·g ∼ Σ (c ∗ d)ₙ einx，其中 (c ∗ d)ₙ = Σₖ cₖ dₙ₋ₖ。这为推导乘积函数的级数提供了系统的方法。
11. From trigonometric to exponential and back | 三角形式与指数形式之间的转换
)/2 allow seamless switching between trigonometric and complex exponential series. Once a series is known in one form, the other form is written immediately, which often clarifies the derivation of a new series.
关系式 aₙ = cₙ + c₋ₙ、bₙ = i(cₙ − c₋ₙ) 以及 cₙ = (aₙ − i bₙ)/2、c₋ₙ = (aₙ + i bₙ)/2 可以在三角形式和复指数形式之间无缝切换。一旦已知一种形式的级数，便可立刻写出另一种形式，这常常使新级数的推导更加清晰。
For example, the full-range sine series of an odd function transforms into a purely imaginary, odd-indexed cₙ sequence. Recognizing this pattern helps to identify possible product series or to apply frequency shifting.
例如，奇函数的全区间正弦级数会转化为纯虚数、奇下标的 cₙ 序列。识别这一模式有助于构造乘积级数或进行频率移位。
12. Worked example: from square wave to triangular wave | 例题：从方波到三角波
1 for −π < x < 0. Its Fourier sine series is (4/π) [ sin x + (1/3) sin 3x + (1/5) sin 5x + … ) yields the triangular wave T(x) = (π/2) − (4/π) [ cos x + (1/9) cos 3x + (1/25) cos 5x + … ], which is even and has the same amplitude pattern but with n² in the denominator.
考虑周期为 2π 的奇方波 S(x)，在 0 < x < π 取 1，在 −π < x < 0 取 −1。其傅里叶正弦级数为 (4/π) [ sin x + (1/3) sin 3x + (1/5) sin 5x + … ]。从 0 到 x 逐项积分（并减去线性项以保证周期性）得到三角波 T(x) = (π/2) − (4/π) [ cos x + (1/9) cos 3x + (1/25) cos 5x + … ]，这是一个偶函数，幅度模式相同但分母为 n²。
Thus, by a single integration step, a completely different Fourier series is obtained. This illustrates how differentiation and integration operations on known series can efficiently generate new expansions without revisiting the integral definitions of the coefficients.
因此，仅通过一步积分，便得到了一个完全不同的傅里叶级数。这展示了如何通过对已知级数进行微分和积分操作，高效生成新的展开式，而无需重新计算系数的积分定义。
Published by TutorHao | IB Mathematics Revision Series | aleveler.com
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