expand f(x,y)=log(x+e^y) in power's of x-1 and y by taylor's theorem
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Answer:
5
To expand a polynomial p(x,y) in powers of x−a and y−b is to find coefficients cjk (which turn out to exist uniquely, and with all but finitely many equal to 0) such that
p(x,y)=∑j,k=0∞cj,k(x−a)j(y−b)k.
(If the total degree of p is N, the sum is limited to indices with j+k≤N.)
One way to achieve this expansion is to write, using your example p(x,y)=xy,
xy=(a+(x−a))(b+(y−b))=ab+b(x−a)+a(y−b)+(x−a)(y−b).(1)
(Analogous use of the binomial theorem handles arbitrary monomials xℓym, and every polynomial in two variables is a finite sum of monomials.)
Another way approach is to invoke Taylor's theorem:
p(x,y)=∑n=0∞∑j=0n∂np∂jx∂n−jy(a,b)(x−a)jj!(y−b)n−j(n−j)!.(2)
Again taking p(x,y)=xy, one finds
c0,0=p(a,b)=ab,c1,0=∂p∂x(a,b)=b,c0,1=∂p∂y(a,b)=a,c1,1=∂2p∂x∂y(a,b)=1,
with all other partial derivatives vanishing, so that (2) agrees with (1). This is presumably the content of Edwards' example.
Generally, (2) yields the coefficients
cj,k=∂j+kp∂jx∂ky(a,b)1j!k!.
There are similar formulas for sufficiently smooth (non-polynomial) functions, perhaps with infinitely many non-zero coefficients, and with "error" or "remainder" terms.
Step-by-step explanation:
answer
expand f(x, y) =log(x+ey) by Taylor's serious in powers of (x-1) & y such that if includes all terms upto second degree