Determination of the Residue Method of Undetermined Coefficient in Partial Fraction
In the process of transforming a rational true fraction into a certain partial fraction sum, the coefficient of the partial fraction can be solved by using the residue.
Here is a paper, the proof can be viewed in the paper.
There is a rational fraction
f (x) = P m (x) Q n (x) f(x)=\frac{P_m(x)}{Q_n(x) }f(x)=Qn(x)Pm(x)
Among them,
m ≤ nm\le nm≤n is
divided into the following two situations for discussion
a. The roots of Q(x)=0 are single roots
. 
That is,The coefficient to be determined at this time is the residue of f(x) at the corresponding pole
b. Nechu Yuju root
The method for determining a single root is the same as that in the case of a. For multiple roots, set x 1 as a multiple root of r, and consider B k
B k = R es [(x − x 1) k − 1 f (x), x 1] When x 1 is (x − x 1) k − 1 f (x ), r − (k − 1) = r − k + 1 pole B_k=Res[(x-x_1)^{k-1)f( x),x_1] \\ At this time x_1 is the r-(k-1)=r-k+1 pole of (x-x_1)^{k-1}f(x)Bk=Res[(x−x1)k−1f(x),x1]This time x1Is ( x−x1)K - . 1 F(X)ofr−(k−1)=r−k+. 1 stage pole point
using the left Poles formula too,
B K =. 1 (R & lt - K) Lim X → X. 1! [(X - X. 1) RF (X)] R & lt - K =. 1 (R & lt - K) ! [(x − x 1) rf (x)] r − k ∣ x = x 1 B_k=\frac{1}{(rk)!}\lim_{x \to x_1} [(x-x_1)^{ r}f(x)]^{rk} \\{\color{Blue}=\frac{1}{(rk)!}[(x-x_1)^rf(x)]^{rk}|_{ x=x_1}}Bk=(r−k)!1x→x1lim[(x−x1)rf(x)]r−k=(r−k)!1[(x−x1)rf(x)]r−k∣x=x1
Note: Although the above conclusions are drawn under real root conditions, after research by bloggers, the above conclusions are also valid when the roots are complex single roots and complex multiple roots.
c. Summary-basic ideas
To convert the rational true fraction f (x) = P m (x) Q n (x) into the form of a partial fraction sum within the real number range, f (x) can be regarded as a special complex variable function f (z ) First transform f (z) into the form of partial fractional sum. According to the integral and residue theory of complex variable function, we can get its undetermined coefficient as f (z ). The residue at the pole must be rational true fraction \\ f(x)=\frac{P_m(x)}{Q_n(x)} \\ in the real number range is reduced to the form of partial fractional sum, f(x) can be regarded as a special complex variable function f(z ) \\First transform f(z) into the form of partial fractional sum, according to the integral and residue theory of complex variable function, we can get \\ whose undetermined coefficient is the residue of f(z) at the pole To be there is reason really points formulaf(x)=Qn(x)Pm(x)In the real number range around the technology as part partial partial formula and the form type , it can be F ( X ) depending of Unexamined special a complex variable function of the number F ( Z )First the F ( Z ) of the as part partial partial formula and the form of formula ,Root According to the complex variable function of the number of plot points and keep the number of physical theory may haveWhich is to be given based the number of F ( Z ) at the very point at the left Number