Hey there! Let's dig into your integral question and break down the possible approaches:

$$ I(k) = \int_{-\infty}^{+\infty} f(g(x)) g'(x) \exp\left(j k x\right) dx \quad (j = \sqrt{-1}) $$

First, let's note that the $f(g(x))g'(x)$ term is a classic chain-rule form—it's the derivative of $F(g(x))$, where $F$ is an antiderivative of $f$. But combining this with the complex exponential $\exp(jkx)$ complicates things, so here are the general strategies to try:

• Start with substitution (the most straightforward first step)
  Let $u = g(x)$, so $du = g'(x)dx$. Rewriting the integral in terms of $u$ gives:
  $$I(k) = \int_{u_{\text{lim}}^{-}}{u_{\text{lim}}^{+}} f(u) \exp\left(j k g^{-1}(u)\right) \cdot \frac{d}{du}g^{-1}(u) du$$
  where $u_{\text{lim}}^{-} = \lim_{x \to -\infty}g(x)$ and $u_{\text{lim}}^{+} = \lim_{x \to +\infty}g(x)$.
  This simplifies nicely if $g(x)$ is an invertible linear function (e.g., $g(x) = ax + b$, $a \neq 0$)—in that case, $g^{-1}(u) = \frac{u - b}{a}$, its derivative is $\frac{1}{a}$, and the integral reduces to a scaled Fourier transform of $f(u)$, which you can compute directly. For non-linear $g(x)$, though, this substitution might just trade one tricky integral for another.

• Check if Fourier transform properties apply (though limited here)
  Standard Fourier transform properties (convolution, frequency shift, time scaling) don't directly handle composite functions like $f(g(x))$. Unless $f(g(x))$ can be decomposed into simpler combinations (e.g., products of basic functions), this route probably won't yield a generic solution. And as you mentioned, expanding these functions into series can be genuinely daunting—so this is likely a last resort.

• Special functions or asymptotic methods for specific cases
  If $g(x)$ has a special form (e.g., exponential, trigonometric), the integral might be expressible using special functions like Bessel functions or error functions. For large $|k|$, the method of stationary phase is a powerful tool for asymptotic approximations: it lets you estimate the integral by focusing on points where the phase $kx$ is "stationary" (where its derivative is zero), avoiding full function expansions.

• Numerical integration as a fallback
  If analytical solutions are out of reach, numerical integration is a practical alternative. You'll want to split the integral into regions where $g(x)$ behaves predictably (e.g., monotonic intervals) and use techniques tailored to oscillatory integrands (like Gaussian quadrature or specialized oscillatory integral methods) to handle the complex exponential term.

I totally get what you mean about those expansions being daunting—they often lead to messy, unmanageable terms, so prioritizing substitution, special cases, or asymptotic methods is way more efficient. If you can share the specific forms of $f$ and $g$, we can dive into more targeted solutions!

备注：内容来源于stack exchange，提问作者CfourPiO