Question

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) ${\left[f\left(x\right)\right]}^3-{\int }_{x_0}^{x}g\left(t\right)dt$ is analytic at x₀.

Short answer

1. Always true.
2. Sometimes False.
3. Always true.
4. Always true

Step-by-step solution

Power series

A power series is an infinite series of the form,$\sum _{n=0}^{\infty }{a}_n{(x-c)}^n=a_0+a_1(x-c)+a_2{(x-c)}^n+.....$

Where, a_nrepresents the coefficient term of the nth term,c is a constant.

Solution for part (a)

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.) .

Solution for part (b)

The given function $\frac{f\left(x\right)}{g\left(x\right)}$.

The above resultant function will hold true only if the denominator $g\left(x\right)\ne 0$, therefore, it may be sometimes false.

Solution for part (c)

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.

Solution for part (d)

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.