Dummy variable in Fourier Transform and Integrals in General

Question

I have a question regarding the use of dummy variables in Fourier Transforms. More specifically, the replacement of a dummy variable of integration inside an integral with another. One of the steps in my textbook required to prove the inverse Fourier transform from the Fourier transform involves a dummy variable such that the following must be true:

$$\psi(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} \phi(k)e^{ikx}dk=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} \phi(u)e^{iux}du$$

IF this is true, one can eventually end up reaching the conclusion that:

$$\phi(k)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\psi(x)e^{-ikx}dx$$

What I don't fully understand is: how come were allowed to simply switch the $k$ for a $u$? I guess a more general question regarding the specific part that I have a doubt with is if, in general, one can switch variables in any integral such that the following is always true:

$$\int_{-\infty}^{\infty}f(k)dk=\int_{-\infty}^{\infty}f(u)du$$

For any $k$ and $u$. Any help would be appreaciated.

Answer 1

You can think of this as a u-substitution, so $k=u \text{ and } dk = du$. But this is solely notation - you're looking at the inverse Fourier transform which is $\phi(x)$, which does not depend at all on the dummy variable.

You can't just swap any $k$ for any $u = p(k)$ without also determining the new differential $du = p'(k) dk$.

Answer 2

Yes, you can always write $\int_{-\infty}^{\infty}f(k)\,dk=\int_{-\infty}^{\infty}f(u)\,du$. Heck, you can even write $\int_{-\infty}^{\infty}f(\ddot{\smile})\,d\ddot{\smile}$, or $\int_{-\infty}^{\infty}f(@)\,d@$ or $\int_{-\infty}^{\infty}f(\sharp)\,d\sharp$. Why? That's just notation.

In general, to talk about integration you need only 2 pieces of information:

• First you need the (nice-enough) function $f$ you're interested in. For now let us focus on functions $f:\Bbb{R}^n\to\Bbb{R}$. Here, let me emphasize that functions are rules: $f$ assigns to each $n$-tuple in $\Bbb{R}^n$ a certain real number. So, these functions are denoted by a single letter $f$. They do not come attached with a variable name like "$f(x)$" or "$f(k)$" or "$f(p)$" or whatnot.
• Next you need the domain of integration, i.e a nice set $E\subset\Bbb{R}^n$.

With these two pieces of information, we can just write the integral as $\int_Ef$; this is the (say RIemann) integral of the function $f$ over the set $E$. That's it. There's no need to bring in auxillary letters $x,u,v,k,t,\xi,\eta,\kappa$ or whatever (sure in hands-on calculations we may like some choices of letters over others; but that's a choice of preference, not dictated by logical necessity. For more about this, you may want to read this answer of mine). This should make sense, because math does not care what your favorite letters are; what is important are the concepts (in this case, integrals, functions and domains of integration).

In single variable calculus, we may be interested in the set $E=[a,b]$ with $a\leq b$, in which case we adopt the notation $\int_a^bf$ in place of $\int_{[a,b]}f$. Similarly, $\int_{-\infty}^{\infty}f$ means $\int_{\Bbb{R}}f$.

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Some side remarks: if you're considering Lebesgue integration, then in $\Bbb{R}^n$ one typically considers the Lebesgue measure, which is typically denoted as $\lambda_n$, or $m_n$, in which case the integrals are written $\int_Ef\,d\lambda_n$ (or $\int_Ef\,dm_n$). Again, there's no necessity for arbitrary letters like $x,u,v,t,k,p...$ (though in practice one sees the notation $\int_Ef(x)\,d\lambda_n(x)$ or simply $\int_Ef(x)\,d^nx$; but again this is for convenience, not out of logical necessity).