(1- 2xt + t?) = (x-t)(x,t) (1) at ag (1- 2xt + t) t g(x,t) (2) ax ag t= (x - t) ag at (3) ere g(x,t) is the generating function of Legendre's polynomials Use Eq. (1) to prove the recurrence relation: (n + 1) Pn+1 = x(2n + 1) P-n Pn-1

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Q: 5. Residue theorem Use the residue theorem to compute (a) ro∞ t2+1 dt.

A: Given integral is∫0∞t2+1t4+1dtThe integral is computed using Residue theorem as shown below.

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Q: 1- Prove the following relations: ag (1 – 2xt + t?)= (x - t) g(x, t) (1) at ag (1- 2xt + t2)= t g(x,…

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Q: If the Legendre polynomials P5 63 x5-70 x3 +15 x then: %3D Find 1) , x* Ps(x) dx 2) P',(-1) A) L*…

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Q: 2. Consider the Newton polynomial Pn(z)= ao+a1 (1-10) + a₂(1-10) (1-11)+... +an(1-10)(1-1₁).…

A: Given that   that interpolate then we have Now  ,       putting

Q: (1+x)

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A: Definition:

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A: By definition, CovaX,bY=abCovX,YCovX+a,Y+b=CovX,Y

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Q: (5) Compute -2 67 using Euler's criterion.

A: Objective: To compute -267. Concept used: Euler's criterion, is a result from number theory that…

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Q: 2 0] Find the characteristic polynomial of A =-1 0 1 0 1 1 2

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Q: 2) For the Fourier series of the function f, ao + Σo acos (kx) + bxsin (kx) we have |bk| ≤ 211/11.…

A: Solution:2)b)FalseWe can solve it by taking a simple exampleFourier series of any function f is a0+…