Mathematical Methods for Physics and Engineering : A Comprehensive Guide

1.2 TRIGONOMETRIC IDENTITIES x y x′ y′ O A B P T N R M Figure 1.2 Illustration of the compound-angle identities. Refer to the ma ...

PRELIMINARY ALGEBRA drawn throughR, the point (0,sin(A+B)) in theOxysystem. That all the angles marked with the symbol•are equal ...

1.2 TRIGONOMETRIC IDENTITIES Consequently the angleθ 12 between two such straight-line graphs is equal to the difference in the ...

PRELIMINARY ALGEBRA and use made of equation (1.15), the following results are obtained: sin 2θ=2sinθcosθ, (1.29) cos 2θ=cos^2 θ ...

1.3 COORDINATE GEOMETRY withKandφas given above. Notice that the inverse sine yields two values in the range 0 to 2πand that the ...

PRELIMINARY ALGEBRA with the coordinate axes, the first three take the forms (x−α)^2 a^2 + (y−β)^2 b^2 = 1 (ellipse), (1.39) (y− ...

1.3 COORDINATE GEOMETRY x y O P F N x=−a (a,0) (x, y) Figure 1.3 Construction of a parabola using the point (a,0) as the focus a ...

PRELIMINARY ALGEBRA the well-known result that the angle subtended by a diameter at any point on a circle is a right angle.
Tak ...

1.4 PARTIAL FRACTIONS in partial fractions, i.e. to write it as f(x)= g(x) h(x) = 4 x+2 x^2 +3x+2 = A 1 (x−α 1 )n^1 + A 2 (x−α 2 ...

PRELIMINARY ALGEBRA numerators on both sides of the equation. This leads to 4 x+2=A 1 (x+2)+A 2 (x+1), 4=A 1 +A 2 2=2A 1 +A 2. S ...

1.4 PARTIAL FRACTIONS Thus any one of the methods listed above shows that 4 x+2 x^2 +3x+2 = − 2 x+1 + 6 x+2 . The best method to ...

PRELIMINARY ALGEBRA We illustrate this procedure with the following worked example.
Find the partial fraction decomposition of ...

1.4 PARTIAL FRACTIONS Aibut linear functions ofx,i.e.oftheformBix+Ci. Thus, in the expansion, linear terms (first-degree polynom ...

PRELIMINARY ALGEBRA In this form, allx-dependence has disappeared from the numerators but at the expense ofp−1 additional terms; ...

1.5 BINOMIAL EXPANSION x=−1 gives respectively 52 24 = C 6 − D 1 2 + 2 4 , 36 7 = B+C 7 −D 1 +2, 86 63 = C−B 7 − D 1 3 + 2 9 . T ...

PRELIMINARY ALGEBRA In fact, the general expression, thebinomial expansionfor powern, is given by (x+y)n= ∑k=n k=0 nC kx n−kyk, ...

1.6 PROPERTIES OF BINOMIAL COEFFICIENTS We start byassumingthat (1.49) is true for some positive integern=N.Wenow proceed to sho ...

PRELIMINARY ALGEBRA The first is a further application of the method of induction. Consider the proposal that, for anyn≥1andk≥0, ...

1.6 PROPERTIES OF BINOMIAL COEFFICIENTS We have specifically included the second equality to emphasise the symmetrical nature of ...

PRELIMINARY ALGEBRA be obvious, but they are both formed in the same way in terms of recurrence relations. Whatever the sign ofn ...