Can I Taylor expand $f(t+a)$ about the variable $t$?

Question

Assume we have a single variable function $f(t)$ and complex numbers $a$ and $\zeta$. The Taylor series of $f(t)$ about $\zeta$ would be: $$ f(t) = f(\zeta) + f^{(1)}(\zeta)(t-\zeta) + \frac{f^{(2)}(\zeta)}{2!}(t-\zeta)^2 + \frac{f^{(3)}(\zeta)}{3!}(t-\zeta)^3 \cdots $$

Now let $t \to t+a$, I am trying to expand $f(t+a)$ about the variable $t$; which would be: $$ f(t+a) = f(t) + f^{(1)}(t)(a) + \frac{f^{(2)}(t)}{2!}(a)^2 + \frac{f^{(3)}(t)}{3!}(a)^3 \cdots + \frac{f^{(n)}(t)}{n!}(a)^n$$ However, I am not sure I can do this? I am asking because $t$ is a variable and not a constant complex number... That being said, doing a Taylor series expansion of $f(t)$ about $t$ would be: $$ f(t) = \sum_{k=0}^\infty \frac{f^{(n)}(t)}{n!}(t-t)^n = f(t)$$ which seems to work...

Answer 1

I suggest you stop thinking in terms of "variables", and stop using the common, but poor language, "a single variable function $f(t)$". The name of the function is $f$; that's it, a single letter is used to denote the function. $f(t)$ is a particular complex number you get by evaluating the function $f$ on the complex number $t\in \text{domain}(f)$. The letters you decide to use does not affect mathematics.

If you want to "Taylor expand $f$ about a complex number $t\in\text{domain}(f)$", then you can write any of the following equivalent statements

for all $a\in\Bbb{C}$ sufficiently close to $0$, $f(t+a)=\sum\limits_{n=0}^{\infty}\frac{f^{(n)}(t)}{n!}a^n$.

or

for all $\sharp \in\Bbb{C}$ sufficiently close to $0$, $f(t+\sharp)=\sum\limits_{n=0}^{\infty}\frac{f^{(n)}(t)}{n!}\sharp^n$.

or

for all $\ddot{\smile} \in\Bbb{C}$ sufficiently close to $0$, $f(t+\ddot{\smile})=\sum\limits_{n=0}^{\infty}\frac{f^{(n)}(t)}{n!}\ddot{\smile}^n$.

or

for all $z \in\Bbb{C}$ sufficiently close to $t$, $f(z)=\sum\limits_{n=0}^{\infty}\frac{f^{(n)}(t)}{n!}(z-t)^n$.

or

for all $w \in\Bbb{C}$ sufficiently close to $t$, $f(w)=\sum\limits_{n=0}^{\infty}\frac{f^{(n)}(t)}{n!}(w-t)^n$.

or

for all $\zeta \in\Bbb{C}$ sufficiently close to $t$, $f(\zeta)=\sum\limits_{n=0}^{\infty}\frac{f^{(n)}(t)}{n!}(\zeta-t)^n$.

I'm going to repeat this once again. The particular letters/symbols used do not matter! What matters is that you're clear on the distinction between a function and its values when evaluated at a point, and that you're clear with what is the operating quantifier in a given statement (eg I always mention "for all blablabla").

If this is still not clear, I suggest you take a look at the slightly related question Why can arbitrary functions be substituted into Taylor Series? where I beat to death the meaning of plugging in something into a function.