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  "body": "\n## Summary\n\nComplex powers $A^s$, $s \\in \\C$, of a positive [[elliptic operator|MT9P2X85V8]] $A$ on a closed manifold generalise the powers $\\lambda^s$ of a positive number.\nSeeley showed that they are [[pseudo-differential operators|FF27N0BWTP]] of order $ms$, where $m$ is the order of $A$, depending holomorphically on $s$.\nTheir traces define the spectral zeta function $\\zeta_A(s) = \\operatorname{tr} A^{-s}$, which extends meromorphically to $\\C$ and is regular at $s = 0$.\nThis is the basis of [[zeta function regularization|VJCJYXGTDY]] of determinants of Laplacians and of analytic torsion.\n\n## Explanations\n\n### L1 human\n\nFor a positive matrix with eigenvalues $\\lambda_j$, the power $A^s$ is the matrix with the same eigenvectors and eigenvalues $\\lambda_j^s$.\nAn elliptic operator such as a Laplacian on a closed manifold has infinitely many eigenvalues tending to infinity, and one can define $A^s$ in the same way.\nFor example, on the circle the operator $1 - d^2/dx^2$ multiplies $e^{inx}$ by $1 + n^2$, and $A^s$ multiplies it by $(1 + n^2)^s$.\nWhen the real part of $s$ is negative enough, these powers are trace class, and their traces, continued to all $s$, give a zeta function attached to the operator.\nIts derivative at $0$ defines a regularised determinant, a finite substitute for the infinite product of the eigenvalues.\n",
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