Fourier Series of a Real Periodic Function

The Fourier series is a mathematical representation of a periodic function as a sum of sine and cosine terms. It allows us to decompose a periodic function into its constituent harmonics. The coefficients of the sine and cosine terms in the Fourier series provide information about the amplitudes and phases of these harmonics.

Even Functions and Fourier Series
An even function is symmetric about the y-axis, which means it is unchanged when reflected across the y-axis. In the context of Fourier series, an even function has the property that its Fourier series representation only contains cosine terms. This is because cosine functions are even functions, and when multiplied with another even function, the result is still an even function.

Odd Functions and Fourier Series
An odd function is symmetric about the origin, which means it is unchanged when reflected across the origin. In the context of Fourier series, an odd function has the property that its Fourier series representation only contains sine terms. This is because sine functions are odd functions, and when multiplied with another odd function, the result is still an odd function.

Correct Statements

P. The Fourier series of a real periodic function has only cosine terms if it is even.
Q. The Fourier series of a real periodic function has only sine terms if it is odd.

Explanation

Statement P: The Fourier series of a real periodic function has only cosine terms if it is even.
This statement is correct. An even function can be represented by a Fourier series consisting of cosine terms only. The cosine terms capture the even symmetry of the function, and the coefficients of these terms provide information about the amplitudes of the even harmonics.

Statement Q: The Fourier series of a real periodic function has only sine terms if it is odd.
This statement is correct. An odd function can be represented by a Fourier series consisting of sine terms only. The sine terms capture the odd symmetry of the function, and the coefficients of these terms provide information about the amplitudes of the odd harmonics.

Conclusion

The correct statements are P and Q. The Fourier series of a real periodic function has only cosine terms if it is even, and only sine terms if it is odd. These properties hold because cosine functions are even functions and sine functions are odd functions. The Fourier series provides a powerful tool for analyzing and representing periodic functions in terms of their constituent harmonics.