# How to compute residues for contour integral of matrix?

I would like to compute $$Y = \int^{\infty}_{-\infty} e^{-itx} (Ix-A)^{-1}dx$$, where $$A$$ is a known square matrix. I am using the semi-circle contour from $$-\infty$$ to $$\infty$$. From the Cauchy integral formula, $$Y = \oint_{semicircle} - \int^{}_{round part}$$.

[EDIT]: as seen on this contour map.

Using $$\oint f(z) (Iz-A)^{-1}dz = 2\pi if(A)$$ for $$|z| > ||A||$$, the round part integral can be evaluated to be $$\frac{2\pi if(A)}{2}$$ from this contour: contour to evaluate outer semi-circle.

However, I am stuck on computing $$\oint_{semicircle}$$ which should be $$\oint f(z) dz = 2\pi i \sum Res$$. How should the residues be computed for matrix contour integrals?

I am aware that the singularities occur at eigenvalues of $$A$$. However, I am clueless about computing them.

• Please clarify your notation a little bit. What exactly do you mean by semi-circle from $-\infty$ to $\infty$, what is $f$ and what is $roundpart$? – weee Nov 9 '18 at 8:33
• @weee I have added a contour map as support – Zetoooo Nov 9 '18 at 13:21

HINT: let any matrix $$B$$ and let $$[B]_{j,k}$$ means the coefficient of row $$j$$ and column $$k$$, then
$$\left[\int f(x)B(x)\, dx\right]_{j,k}=\int f(x)[B(x)]_{j,k}\, dx$$
So you can evaluate the integral coefficient by coefficient of the matrix $$B$$. Also remember that
$$G\int f(x)\, dx=\int Gf(x)\, dx$$ for any bounded linear operator $$G$$ (by example, a square matrix). So you can use the matrix $$(Ix-A)^{-1}$$ in Jordan form if you please (or any other form easy to manipulate) and after change it back to it standard basis.