Fourier Series of f(x) cos(ax) and its Complex Form

In this article, we will explore the Fourier series representation of the function f(x) cos(ax). The Fourier series is a powerful tool for representing periodic functions as a sum of simpler, trigonometric functions. We will focus on the complex form of the Fourier series, which allows us to express the function in terms of exponentials.

Introduction to Fourier Series

Fourier series is a method to represent any periodic function as an infinite sum of sine and cosine functions. For a periodic function f(x) with period T 2π, the Fourier series can be expressed in its complex form as:

f(x) Σn-∞∞ γne2iπnx/T

Where the coefficients γn are given by:

γn (1/T) ∫-T/2T/2 f(x)e-2iπnx/T dx

The Complex Form of f(x) cos(ax)

Let's consider the function f(x) cos(ax), where a is a constant. We extend this function with period 2π to the entire x domain (-∞, ∞).

The coefficients γn for the function can be calculated as:

γn (1/2π) ∫-ππ cos(ax) e-inx dx

Let's break down the integral and solve for γn:

γn (1/2π) ∫-ππ cos(ax) e-inx dx

Using the integration by parts or the property of the exponential function, we get:

γn (1/2π) [ (e-inx / (a^2 - n^2) e-inx - in cos(ax) a sin(ax)) |-ππ ]

Simplifying:

γn (1/2π) [ (1 / (a^2 - n^2) [ e-inπ - in cos(aπ) sin(aπ) - einπ - in cos(aπ) sin(aπ) ] ]

Further simplifying:

γn (1/2π) [ (1 / (a^2 - n^2) [ asin(aπ) einπ e-inπ - in cos(aπ) - einπ - in cos(aπ) ] ]

This reduces to:

γn (1/π) [ -1n / (a^2 - n^2) asin(aπ) ]

Thus, the complex form of the Fourier series for f(x) cos(ax) is:

f(x) Σn-∞∞ (1/π) [-1n / (a^2 - n^2) asin(aπ)] einx

Proof that the Result is a Real Function

To prove that the Fourier series of f(x) cos(ax) is a real function, we need to show that the imaginary part is zero. The series can be expanded and simplified as:

f(x) (1/π) a sin(πa) Σn0∞ (-1n) / (a2 - n2) cos(nx)

Since only even terms are present, the series simplifies further to:

f(x) (2/π) a sin(πa) Σn1∞ [1 / (a2 - n2) (1 / (a2 - n2) - 1 / (a2 - (2n)2) cos(nx)

Which can be written as:

f(x) (2/π) a sin(πa) [1 / (a2 - 1) cos(x) 1 / (a2 - 4) cos(2x) 1 / (a2 - 9) cos(3x) ... ]

This confirms that the Fourier series of f(x) cos(ax) is indeed a real function.

Conclusion

The Fourier series of the function f(x) cos(ax) has been derived and expressed in its complex form. The result shows that the Fourier series is a real function and can be represented as a sum of cosine terms with varying amplitudes.

Understanding the Fourier series and its complex form is crucial for analyzing periodic phenomena in various fields such as physics, engineering, and signal processing. The derived series provides a powerful tool for decomposing complex periodic functions into simpler sine and cosine components.