The Residue Theorem Behind Partial Fractions

The Residue Theorem Behind Partial Fractions#

Several times now we have broken a rational function of the lag operator into a sum of first-order pieces of the form \(\dfrac{A_j}{1-\lambda_j L}\). We did it to invert a moving average operator, writing \(d(L)^{-1} = \sum_j A_j/(1-\lambda_j L)\) in (258); we will do it again, implicitly, every time we apply the annihilation operator \((\cdot)_+\) of the Wiener–Kolmogorov prediction formula (248). This “partial-fractions” maneuver is not a bag of tricks. Its coefficients are residues, and the legitimacy of operating on the pieces one at a time rests on the residue theorem and the Laurent expansions of A Mathematical Digression: Fourier and z Transforms and Evaluating the Inverse z Transform. This short section makes the connection explicit and shows why first-order building blocks keep reappearing.

Partial-fraction coefficients are residues#

Let \(g(z) = N(z)/D(z)\) be a ratio of polynomials with \(\deg N < \deg D\) (if not, divide out the polynomial part first), and suppose \(D\) has distinct zeros \(p_1,\dots,p_n\). Then \(g\) has a partial-fraction expansion

(265)#\[g(z) = \sum_{j=1}^{n} \frac{r_j}{z - p_j}, \qquad r_j = \operatorname*{Res}_{z=p_j} g(z) = \lim_{z\to p_j}(z-p_j)\,g(z) = \frac{N(p_j)}{D'(p_j)}.\]

The coefficient \(r_j\) is exactly the residue of \(g\) at the simple pole \(p_j\) defined in (178). The “cover-up rule” of elementary calculus — multiply by \((z-p_j)\), set \(z=p_j\) — is the residue evaluation. The residue theorem supplies the reason the two agree: integrating (265) around a small circle \(\gamma_j\) enclosing only \(p_j\),

(266)#\[\frac{1}{2\pi i}\oint_{\gamma_j} g(z)\,dz = r_j,\]

since each term \(\frac{r_k}{z-p_k}\) integrates to \(r_k\) if \(p_k\) is inside \(\gamma_j\) and to \(0\) otherwise. The residue is at once the coefficient in the expansion and the contour integral; that is what lets us pass between the two views.

For time-series work it is more convenient to factor the denominator into the lag-operator-friendly form \(D(z) = \prod_{j}(1-\lambda_j z)\), whose poles sit at \(z = 1/\lambda_j\). The expansion becomes

(267)#\[g(z) = \frac{N(z)}{\prod_{j=1}^{n}(1-\lambda_j z)} = \sum_{j=1}^{n} \frac{A_j}{1-\lambda_j z}, \qquad A_j = \Big[(1-\lambda_j z)\,g(z)\Big]_{z=1/\lambda_j} = \frac{N(1/\lambda_j)}{\prod_{k\neq j}\big(1-\lambda_k/\lambda_j\big)} .\]

This \(A_j\) is the same residue in different clothing: because \(1-\lambda_j z = -\lambda_j(z - 1/\lambda_j)\), the two coefficients are related by \(A_j = -\lambda_j\, \operatorname*{Res}_{z=1/\lambda_j} g(z)\). The cover-up rule (267) is the form we use in practice.

Repeated poles. When \(D\) has a zero of order \(m\), say a factor \((1-\lambda z)^m\), the expansion carries terms \(\dfrac{A^{(1)}}{1-\lambda z} + \cdots + \dfrac{A^{(m)}}{(1-\lambda z)^m}\), and the coefficients are the higher-order residues of (177), obtained by differentiating \((1-\lambda z)^m g(z)\). The order-\(m\) residue formula of Evaluating the Inverse z Transform is precisely what a repeated pole demands.

Why this is the natural tool: inversion pole by pole#

The payoff is that each simple term is the \(z\)-transform of a geometric sequence. For \(|\lambda_j|<1\),

(268)#\[\frac{A_j}{1-\lambda_j z} = A_j\sum_{k=0}^{\infty}\lambda_j^{k} z^{k} \quad\Longleftrightarrow\quad \{A_j\lambda_j^{k}\}_{k\ge 0},\]

one-sided in nonnegative powers — exactly Example 1 of Evaluating the Inverse z Transform, computed there by residues. A partial-fraction expansion therefore turns the inversion of a complicated rational function into the inversion of a handful of geometrics, one per pole, which are then added. This is the same calculation as the inverse-\(z\)-transform-by-residues of (179), simply organized by pole rather than performed as a single contour integral.

Two facts we have already used are instances of it.

  • Inverting an \(m\)th-order moving average. The expansion \(d(L)^{-1}=\sum_j A_j/(1-\lambda_j L)\) of (258) gives the autoregressive weights as a sum of geometrics, \(a_k = \sum_j A_j\lambda_j^{k}\). Convergence of (259) is just the requirement \(|\lambda_j|\le 1\) that makes each geometric (268) summable — the fundamental-versus-nonfundamental distinction read off the poles.

  • The autocovariances of an autoregression. The closed form (181) for \(c_y(\tau)\) — a sum over the poles \(\lambda_j\) of residue terms — is nothing but the partial-fraction expansion of the covariance generating function \(g_y(z)\) inverted term by term. The “sum of residues inside \(\Gamma\)” of (180) and the partial-fraction sum are the same object.

Example 1: inverting a moving average operator#

Let \(d(L) = (1-\tfrac12 L)(1-\tfrac13 L)\), so that \(\lambda_1=\tfrac12,\ \lambda_2=\tfrac13\), and we want \(d(L)^{-1} = \sum_j A_j/(1-\lambda_j L)\). The cover-up rule (267) with \(N\equiv 1\) gives

\[ A_1 = \frac{1}{1-\lambda_2/\lambda_1} = \frac{1}{1-\tfrac{2}{3}} = 3, \qquad A_2 = \frac{1}{1-\lambda_1/\lambda_2} = \frac{1}{1-\tfrac{3}{2}} = -2 , \]

so that

\[ d(L)^{-1} = \frac{3}{1-\tfrac12 L} - \frac{2}{1-\tfrac13 L}, \qquad a_k = 3\left(\tfrac12\right)^{k} - 2\left(\tfrac13\right)^{k}. \]

As a check, \(a_0 = 3-2 = 1\) (consistent with \(d_0=1\)), and \(a_1 = \tfrac32 - \tfrac23 = \tfrac56\), which matches the coefficient of \(L\) in the direct inverse of \(d(L) = 1 - \tfrac56 L + \tfrac16 L^2\). The weights decay at the rate of the largest pole reciprocal, \(\lambda_1=\tfrac12\), because that geometric dominates the tail — a fact that the partial-fraction form displays at a glance but the recursive inversion hides.

If instead the pole is repeated, \(d(L) = (1-\tfrac12 L)^2\), the order-two residue (177) gives \(d(L)^{-1} = \sum_{k\ge 0}(k+1)\left(\tfrac12\right)^{k} L^{k}\): a geometric multiplied by the polynomial \(k+1\), the signature of a double pole.

Example 2: the annihilation operator is diagonal in this basis#

The prediction formula (248) hinges on the annihilation operator \((\cdot)_+\), which discards negative powers of \(L\). Partial fractions is what makes it computable, because \((\cdot)_+\) acts on each first-order piece independently. Consider the mixed process of Some Examples,

\[ x_t = \frac{1+aL}{1-\beta L}\,\epsilon_t, \qquad |a|<1,\ |\beta|<1, \]

with \(\epsilon_t\) fundamental. To form \(P_{t-1}x_t = \big(\tfrac{d(L)}{L}\big)_+ d(L)^{-1} x_{t-1}\) we expand the inside operator in partial fractions before annihilating:

\[ \frac{L^{-1}(1+aL)}{1-\beta L} = \underbrace{\frac{L^{-1}}{1-\beta L}}_{\text{has an }L^{-1}\text{ term}} + \frac{a}{1-\beta L}. \]

The second piece is already a sum of nonnegative powers and survives untouched; the first contributes its negative power \(L^{-1}\) (annihilated) and leaves the geometric tail \(\beta/(1-\beta L)\). Hence

\[ \left(\frac{L^{-1}(1+aL)}{1-\beta L}\right)_+ = \frac{\beta + a}{1-\beta L}, \qquad P_{t-1}x_t = \frac{\beta+a}{1-\beta L}\cdot\frac{1-\beta L}{1+aL}\,x_{t-1} = \frac{a+\beta}{1+aL}\,x_{t-1}, \]

reproducing the result of Some Examples. The lesson is structural: \((\cdot)_+\), like inversion and like filtering, is diagonal in the partial-fraction basis — it can be applied to one pole at a time. That is the real reason the first-order operator \(1/(1-\lambda L)\) is the recurring atom of this theory.

Echoes and influences#

This circle of ideas — factor, expand into one term per pole, operate pole by pole, add — runs through the neighboring sections:

A unifying way to see it: partial fractions diagonalizes a rational lag operator into a sum of one-pole operators, in the same way that an eigen-decomposition diagonalizes a matrix. Operations that look formidable on the original operator — inverting it, predicting with it, filtering through it, annihilating its negative powers — become trivial on each \(1/(1-\lambda L)\) piece and are simply summed back. The residues are the weights in that decomposition, and the residue theorem is the guarantee that the pieces fit back together.

Exercises#

Exercise 28 (Residues, inversion, and a repeated pole)

  1. Let \(d(L) = (1-0.6L)(1-0.2L)\). Use the cover-up rule (267) to find \(A_1,A_2\) in \(d(L)^{-1} = A_1/(1-0.6L) + A_2/(1-0.2L)\), and write the autoregressive weights \(a_k\). Verify that \(a_0 = 1\) and that \(a_1\) equals the coefficient of \(L\) in the direct expansion of \(d(L)^{-1}\). Which pole governs the rate at which \(a_k\to 0\)?

  2. Confirm the residue/cover-up coefficient \(A_j\) equals \(-\lambda_j\) times the ordinary residue \(\operatorname*{Res}_{z=1/\lambda_j} d(z)^{-1}\) of (178), for this \(d\).

  3. Now take the repeated pole \(d(L) = (1-0.6L)^2\). Using the order-two residue (177), show \(d(L)^{-1} = \sum_{k\ge0}(k+1)(0.6)^k L^k\), and explain in one line why a double pole produces a geometric multiplied by a polynomial in \(k\).

Exercise 29 (Annihilation pole by pole)

Let \(x_t = \dfrac{1}{(1-\tfrac12 L)(1-\tfrac13 L)}\,\epsilon_t\) with \(\epsilon_t\) fundamental white noise.

  1. Expand \(\dfrac{1}{(1-\tfrac12 L)(1-\tfrac13 L)}\) in partial fractions (you computed the coefficients in Example 1) and write the moving-average weights \(d_k\).

  2. Use the Wiener–Kolmogorov formula (248) with \(k=1\), \(P_{t-1}x_t = \big(\tfrac{d(L)}{L}\big)_+\, d(L)^{-1}\,x_{t-1}\), applying \((\cdot)_+\) to each partial-fraction term separately, to show that the one-step-ahead forecast is a geometric distributed lag of past \(x\)’s. Identify its decay rate.

  3. Optional. Compute the autocovariances \(c_x(\tau)\) from the residue formula (181) and confirm that they are a sum of two geometrics in \(|\tau|\), one per pole — the covariance-generating-function counterpart of the partial-fraction expansion in part 1.

References#

[PFCBV74]

Ruel V. Churchill, James W. Brown, and Roger F. Verhey. Complex Variables and Applications. McGraw-Hill, New York, 3 edition, 1974.

[PFHS80]

Lars Peter Hansen and Thomas J. Sargent. Formulating and estimating dynamic linear rational expectations models. Journal of Economic Dynamics and Control, 2(1):7–46, 1980.

[PFWhi83]

Peter Whittle. Prediction and Regulation by Linear Least-Square Methods. University of Minnesota Press, Minneapolis, 2 edition, 1983.