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Building on an insight due to Avramidi, we provide a system of transport equations for determining key fundamental bitensors, including derivatives of the world function, ðx; x0Þ, the square root of the Van Vleck determinant, 1=2ðx; x0Þ, and the tail term, Vðx; x0Þ, appearing in the Hadamard form of the Green function. These bitensors are central to a broad range of problems from radiation reaction to quantum field theory in curved spacetime and quantum gravity. Their transport equations may be used either in a semi-recursive approach to determining their covariant Taylor series expansions, or as the basis of numerical calculations. To illustrate the power of the semi-recursive approach, we present an implementation in MATHEMATICA, which computes very high order covariant series expansions of these objects. Using this code, a moderate laptop can, for example, calculate the coincidence limit ½a7ðx; xÞ and Vðx; x0Þ to order ðaÞ20 in a matter of minutes. Results may be output in either a compact notation or in XTENSOR form. In a second application of the approach, we present a scheme for numerically integrating the transport equations as a system of coupled ordinary differential equations. As an example application of the scheme, we integrate along null geodesics to solve for Vðx; x0Þ in Nariai and Schwarzschild spacetimes.
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