Advanced Mathematical Methods in Science and Engineering by S.I. Hayek

By S.I. Hayek

A set of an intensive variety of mathematical subject matters right into a plenary reference/textbook for fixing mathematical and engineering difficulties. issues coated contain asymptotic equipment, a proof of Green's capabilities for traditional and partial differential equations for unbounded and bounded media, and extra.

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Since the characteristic equation in eq. ~)(0. - ¢rz) then multiplying eq. :~) = ao x~’-2 0. 1 0. (0")x n=0 n=0 n=k so that the coefficient ak is the first term of the secondseries. Differentiating the expressionas given in eq. 19) one obtains: CHAPTER 2 34 n+O E(o-~2)anCo)xn+° ~--~[(~- if2) Y(X,~)] n=O n=k-1 n+ff+ =logx, + E(c~-c)2)an(O)X n=k n=k-1 E(ff-ff2)an(ff)xn+ff+ n=k--1 ~(ff-ff2)a~(ff)x n=0 E an(~) n=0 n=0 xn+° ~+~ +logx + 2[(~-~2)a~(ff)l n=k n=k It should be no~d~t ~(ff) = - (gn(ff))/(ffff+n)) does not con~n~e te~ (if’if2) denominatoruntil n = k, ~us: ~d for n=0, 1,2 .....

N. 1) n=O where c. : Y I series diverges n-~ ~ nCn(X-X0) < p series converges ¯ Ix - x°l > p series diverges where p is known as the Radius of Convergence.

30 x4 x7 Y2 = x+--+~+... 12 12-42 are the two independentsolutions of the homogeneous differential equation. 1, Cn_2 ..... Cl, c o in a formula knownas the Recurrence Formula. : 0"c0x-2+0"Clx-l+2c2 + n(n-1)Cn xn-2- n=3 1. wherethe first term of each powerseries starts with x n+l ~’~ ~ C nX n=0 =0 CHAPTER 2 22 Letting n = m+ 3 in the first series and n = min the secondseries, so that the two series start with the sameindex m= 0 and the powerof x is the samefor both series, one obtains: c2 =: 0 Co = indeterminate Cl = indeterminate Z[(m+ 2)(m+ 3)Cm+3--Cm]X m+l :0 m=0 Equatingthe coefficient of x’~’1 to zero gives the recurrenceformula: Cm = m= 0, 1, 2 ....

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