Math Problem Statement

Find the Fourier transform of fx={ 1-x ; x<1 0 ; x>1

Hence evaluate 0(sintt)4dt.

Solution

Let's go through the process step-by-step.

Part 1: Fourier Transform of f(x)=1−xf(x) = 1 - x for x<1x < 1 and f(x)=0f(x) = 0 for x>1x > 1

We are asked to find the Fourier transform of a function f(x)f(x) defined as:

\begin{cases} 1 - x, & x < 1, \\ 0, & x > 1 \end{cases}$$ The Fourier transform $$F(k)$$ of $$f(x)$$ is given by: $$F(k) = \int_{-\infty}^{\infty} f(x) e^{-ikx} \, dx$$ Since $$f(x) = 0$$ for $$x > 1$$, the limits of the integral are effectively from $$-\infty$$ to 1. But $$f(x)$$ is zero for $$x > 1$$, so we only need to consider the interval from $$-\infty$$ to 1. Also, for $$x < 1$$, $$f(x) = 1 - x$$. So the integral becomes: $$F(k) = \int_{-\infty}^{1} (1 - x) e^{-ikx} \, dx$$ Now, we can split the integral: $$F(k) = \int_{-\infty}^{1} e^{-ikx} \, dx - \int_{-\infty}^{1} x e^{-ikx} \, dx$$ ### Part 2: Evaluating the Integrals We will evaluate each of the integrals separately. 1. **First Integral:** $$I_1 = \int_{-\infty}^{1} e^{-ikx} \, dx$$ This is a standard integral: \

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Math Problem Analysis

Mathematical Concepts

Fourier Transform
Integral Calculus

Formulas

Fourier transform: F(k) = ∫ f(x) e^{-ikx} dx
Standard integral of exponential function: ∫ e^{-ikx} dx

Theorems

Fourier Transform Theorem
Standard Integration Theorems

Suitable Grade Level

University Level - Undergraduate

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