By Relton, Frederick Ernest

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Ramakrishna Ramaswamy's D.D. Kosambi: Selected Works in Mathematics and Statistics PDF

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Additional info for Applied Bessel functions.

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A) 36 APPLIED BESSEL FUNCTIONS which has consecutive zeros at a, a + 7ijk and satisfies the equation w" + k2w = 0. Then f (k2 — I)vwdx =Jf Ja (wvh a Take ¡3 to be the zero o f v next following a and suppose that, if possible, it falls short of a + 7t/L This ensures that (vx), w(x) and k2 — I(x) are all positive (or at least, not negative) between a and ¡3. The integral is therefore positive; moreover, v(P) = 0 = v(a) = w( a). The right side accordingly reduces to w(/3)v'(f3) where v'(f}) is negative.

Jo The integral is zero if a, p are zeros of aJn(t) + btJn'(t); and when else? x)Jn+1(a#)}. 17. If m + n > 0, prove that f o \ Jm(«x)Jn(*x) dx = {J n(a )J ,/(a ) - J n'(a )J 'm(«)}. Deduce by a limiting process that So i J» ^ dx= £ j »2(a>£ 18. Jz(x) + ^ (z 2 — 16)J4(a;). 19. Establish the reduction formula J x m+LJndx (m2 — n2)J x m~1Jndx = xm+lJn+1 -f (m — ri)xmJn. 20. Prove that the solution of xy" + \y' + \y = 0 is x*{AJ±(V x) + BJ_\$(Vx)}. 21. Find the general solution of the equation * y - 2xy' + 4CX4 - l)y = 0.

The formal method o f obtaining the series is to assume y = a0xT+ axxr+1 + a2xr+2 + . . I f this, with its derivatives, be substituted in the differential equation, the result should be identically zero, so that the coefficient o f every power of x vanishes. This procedure supplies a set o f recurrence for­ mulae whereby all the coefficients can be determined in terms of the first, which remains arbitrary. The very first coefficient supplies an equation, known as the “ indicial equation ” , which decides the per­ missible values of the leading index r.