Question: Let f(x) and g(x) be analytic at x0. Determine whether the following statements are always true or sometimes false:
(a) 3f(x)+g(x) is analytic at x0 .
(b) f(x)/g(x) is analytic at x0 .
(c) f'(x) is analytic at x0 .
(d) is analytic at x0 .
A power series is an infinite series of the form,
Where, an represents the coefficient term of the nth term,c is a constant.
It is given that the functions f(x) and g(x) are both analytic at x=x0,therefore both the functions can be written in the form of power series and are convergent in the vicinity of x0
The given function 3f(x)+g(x
The above linear combination of the analytic functions will always hold true because the resultant function will also have a power series representation, this follows directly from the property of summation.) .
The given function .
The above resultant function will hold true only if the denominator , therefore, it may be sometimes false.
The given function f'(x).
The derivative of the analytic function will also be analytic, as the analytic function have every order of derivative at the point x0 therefore the derivatives of analytic functions will also be analytic. Thus this will always be true.
The given function
The above combination of analytic functions will always be analytic, as the operations performed on the functions result in analytic functions. The function f(x) is multiplied three times which will give another analytic function and the function g(x) is integrated, the term-wise integration of the analytic function will also yield another analytic function.
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