(1)
then the functions are known as analytic extensions of each other. In particular, is the analytic extension of in the rest of the domain of ; and is the analytic extension of in the rest of the domain of .
In certain applications, like in the evaluation of divergent integrals, analytic extension theory is used to assign values to functions outside their domains. So, for example, one can use to compute the value of the function for ‘s outside but falling in , and vice versa. However, that is not the application we have in mind in this post.
What we have in mind is the following. Let us say we are given two analytic functions and with intersecting domains, and , with the intersection given by . Let us say that the functional forms of the functions are so different that it is not immediately clear whether they are equal in their common domain or not. But let us say that we have established that there exists a segment, , contained in , such that the functions are equal there, for all .
The question relevant to us in this post is Given the information on the equality of the functions on the segment, what can we conclude on the values of the functions on their common domain? Analytic extension theory says that equality of two analytic functions on a segment contained in their common domain implies their equality in the rest of their common domain. Explicitly
(2)
This is a powerful result that is a must learn for every student of mathematical physics.
and D_1D_2D_{12}\GammaD_12f_1(z)f_2(z)D_1D_2D_{12}\GammaD_{12}
f(t)\mathrm{Re}(z)>0F(z)D=\{z\in\mathbb{C}| \mathrm{Re}(z)>0\}F(z)z=x+ i yx>0y\in\mathbb{R}
F_c(x,y)F_s(x,y)F(z)
xyF_cF_sxyF_c(x,y)yxF_c(x,y)\sigma
\sigma\mathrm{Re}(\sigma)>0\tilde{F}_c(\sigma,y)F_c(x,y)\tilde{F}_c(\sigma,y)\sigma\tilde{F}_c(\sigma,y)F_c(x,y)
\sigma\mathrm{Re}(\sigma)>0\tilde{F}_c(\sigma,y)\tilde{F}_c'(\sigma,y)\tilde{F}_c(\sigma,y)\tilde{F}_c'(\sigma,y)\sigma>0\tilde{F}_c(\sigma,y)=\tilde{F}_c'(\sigma,y)\sigma\mathrm{Re}(\sigma)>0y
\sigma\mathrm{Re}(\sigma)>0y\in \mathbb{R}F_s(x,y)
\sigma\mathrm{Re}(\sigma)>0y\in \mathbb{R}\sigma\sigma\sigmay
\gamma\sigma$, given the known integral
(14)
Finally, analytic extension theory offers a powerful way of solving problems which take the complex plane as its domain. We have seen that it is possible to reduce the problem to a segment in the relevant domain and then obtain the full solution by analytic extension.