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A Course in Mathematical Analysis - download pdf or read online

By Goursat E.

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Extra info for A Course in Mathematical Analysis

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Start at (x z, , we have y Fm 4 with the value ) a definite value of this root as long as we remain Let 1 (x l y^ be a point of the path inside R, M inside the region R. and ~ , z t the corresponding value of x =x y=y The z. =z conditions of the theorem there exists another region lt v l} being R l} about the point MI, inside which the root which reduces to z l for satisfied for x = Xi, y = yi is z , This uniquely determined. new region #! will M Taking then such a point t but outside R, we may repeat the same con have, in general, points outside of R.

XXII. lemma. TRANS. IMPLICIT FUNCTIONS 25] II, which for is satisfied x =x du and from equation y , ^ y , t = zw z 47 =u u Now . /), ; x , is replaced is equal to (x, y, z) in continuous function. for v also a certain The proposition is then proved for a system of two equations. can show, as in 21, that these functions possess partial derivatives of the first order. Keeping y and z constant, let us We give x an increment Ax, and let AM and Ay be the corresponding increments of the functions u and v.

Another infinitesimal, and ft son, and be the principal said to be an is is infinitesimal of higher order with respect to a, if the ratio ft/a On the other hand, ft is called an infini zero with a. approaches of tesimal the order with respect to a, first approaches a limit this case K different ^ where =K+ ft=a(K + c)= Ka if a. Hence + at, Ka The complementary term is called the principal part of ft. an infinitesimal of higher order with respect to a. In general, such that ft we can find a positive power of a, say and at the ratio ft/a In zero.

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A Course in Mathematical Analysis by Goursat E.

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