- #1
JBD
- 15
- 1
I'm trying to expand the following using Newton's Generalized Binomial Theorem.
$$[f_1(x)+f_2(x)]^\delta = (f_1(x))^\delta + \delta (f_1(x))^{\delta-1}f_2(x) + \frac{\delta(\delta-1)}{2!}(f_1(x))^{\delta-2}(f_2(x))^2 + ...$$
where $$0<\delta<<1$$
But the condition for this formula is that $$\lvert f_1(x)\rvert > \lvert f_2(x)\rvert$$
And that's where my problem is. Since both functions are sinusoidal, there are times when indeed $$\lvert f_1(x)\rvert > \lvert f_2(x)\rvert$$ but there are also values of x such that $$\lvert f_2(x)\rvert > \lvert f_1(x)\rvert$$. Take for example the graphs of cos^2 x and sin^2x.
In other words, since the condition is violated, the expansion is not true for all x.
I'm thinking of separating the two instances. At x's where $$\lvert f_1(x)\rvert > \lvert f_2(x)\rvert$$ then I can use the above expansion. If $$\lvert f_2(x)\rvert > \lvert f_1(x)\rvert$$, then:
$$[f_2(x)+f_1(x)]^\delta = (f_2(x))^\delta + \delta (f_2(x))^{\delta-1}f_1(x) + \frac{\delta(\delta-1)}{2!}(f_2(x))^{\delta-2}(f_1(x))^2 + ...$$
But, how can I separate the two instances? Or is there another way to solve this problem?
$$[f_1(x)+f_2(x)]^\delta = (f_1(x))^\delta + \delta (f_1(x))^{\delta-1}f_2(x) + \frac{\delta(\delta-1)}{2!}(f_1(x))^{\delta-2}(f_2(x))^2 + ...$$
where $$0<\delta<<1$$
But the condition for this formula is that $$\lvert f_1(x)\rvert > \lvert f_2(x)\rvert$$
And that's where my problem is. Since both functions are sinusoidal, there are times when indeed $$\lvert f_1(x)\rvert > \lvert f_2(x)\rvert$$ but there are also values of x such that $$\lvert f_2(x)\rvert > \lvert f_1(x)\rvert$$. Take for example the graphs of cos^2 x and sin^2x.
In other words, since the condition is violated, the expansion is not true for all x.
I'm thinking of separating the two instances. At x's where $$\lvert f_1(x)\rvert > \lvert f_2(x)\rvert$$ then I can use the above expansion. If $$\lvert f_2(x)\rvert > \lvert f_1(x)\rvert$$, then:
$$[f_2(x)+f_1(x)]^\delta = (f_2(x))^\delta + \delta (f_2(x))^{\delta-1}f_1(x) + \frac{\delta(\delta-1)}{2!}(f_2(x))^{\delta-2}(f_1(x))^2 + ...$$
But, how can I separate the two instances? Or is there another way to solve this problem?