[ \mathcal{F}[f(t)] = \int_{0}^{\infty} e^{-2t} \sin t \cdot e^{-i \omega t} , dt = \int_{0}^{\infty} e^{-(2 + i \omega)t} \sin t , dt ]写成公式

当然，可以将傅里叶变换的步骤写成公式的形式：

\[ \mathcal{F}[f(t)] = \int_{0}^{\infty} e^{-2t} \sin t \cdot e^{-i \omega t} \, dt = \int_{0}^{\infty} e^{-(2 + i \omega)t} \sin t \, dt \]

其中，\( f(t) = e^{-2t} u(t) \sin t \)，且 \( u(t) \) 是单位阶跃函数，所以积分从 \( 0 \) 到 \( \infty \) 进行。

如有其他问题或需要进一步解释，请告诉我。

Learn how to compute the Fourier Transform of a function involving exponential decay and sine functions. This problem demonstrates the application of Fourier Transform formula and relates to concepts like exponential decay and sine functions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation