Which of the following statements is true?

Explanation:

Trigonometric Fourier Series of Even and Odd Functions:

The Fourier series is a mathematical tool used to represent periodic functions as a sum of sines and cosines (or exponentials in the complex form). The trigonometric Fourier series is particularly helpful for analyzing real-valued periodic signals and decomposing them into their frequency components.

Key Characteristics of Even and Odd Functions:

• An even function satisfies the property f(-t) = f(t) . Graphically, even functions are symmetric about the vertical axis.
• An odd function satisfies the property f(-t) = -f(t). Graphically, odd functions are symmetric about the origin.

Trigonometric Fourier Series Properties for Even and Odd Functions:

• Even Functions: The Fourier series of an even function contains only the DC term and cosine terms. This is because the sine terms, which are odd functions, vanish due to the symmetry of the even function.
• Odd Functions: The Fourier series of an odd function contains only sine terms. The DC term and cosine terms vanish due to the symmetry of the odd function.

Correct Option Analysis:

The correct option is:

Option 4: Trigonometric Fourier series of an even function of time contains only DC term and cosine terms.

This option is correct because, for an even function, the sine terms disappear due to the function's symmetry. The Fourier series representation of an even function is expressed as:

\(f(t) = a_0 + \sum_{n=1}^{\infty} a_n \cos(n\omega t)\)

Here, \( a_0 \) represents the DC (average) term, and \( a_n \) represents the coefficients of the cosine terms. Since the function is even, all sine terms \( b_n \sin(n\omega t) \) vanish.

Mathematical Proof:

For an even function \( f(t) \), the Fourier coefficients are derived as:

• \( a_0 = \frac{1}{T} \int_{-T/2}^{T/2} f(t) dt \) (DC term)
• a_n = \(\frac{2}{T} \int_{0}^{T/2} f(t) \cos(n\omega t) dt \) (cosine coefficients)
• b_n = \(\frac{2}{T} \int_{0}^{T/2} f(t) \sin(n\omega t) dt \) = 0  (sine coefficients vanish due to the even symmetry)

Thus, the Fourier series for even functions contains only the DC term and cosine terms