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Extra info for 4-Manifold topology II: Dwyers filtration and surgery kernels
5). 5 Power series, two variables The idea of a power series can be extended to more than one variable. One way to develop it is to use exactly the same sort of brute-force approach that I used for the one-variable case. Assume that there is some sort of infinite series and successively evaluate its terms. f (x, y ) = A + Bx + Cy + Dx2 + Exy + F y 2 + Gx3 + Hx2 y + Ixy 2 + Jy 3 · · · Include all the possible linear, quadratic, cubic, and higher order combinations. Just as with the single variable, evaluate it at the origin, the point (0, 0).
This is not a handicap, as there are many circumstances for which this is all you need. I derived this assuming that n is large, but notice that the result is not too bad even for modest values. The error is less than 2% for n = 5. There are even some applications, especially in statistical mechanics, in which you can √ make a still cruder approximation and drop the factor 2πn. That is because in that context it is the logarithm of n! that appears, and the ratio of the logarithms of the exact and approximate numbers goes to one for large n even without this square root factor.
The double factorial of an even integer can be expressed in terms of the single factorial. Do so. What about odd integers? 20 Evaluate this integral. Just find the right substitution. 21 A triangle has sides a, b, c, and the angle opposite c is γ. Express the area of the triangle in terms of a, b, and γ. Write the law of cosines for this triangle and then use sin2 γ + cos2 γ = 1 to express the area of a triangle solely in terms of the lengths of its three sides. The resulting formula is not especially pretty or even clearly symmetrical in the sides, but if you introduce the semiperimeter, s = (a + b + c)/2, you can rearrange the answer into a neat, symmetrical form.