Question

If \(f(x)=\frac{1}{2} a_{0}+\sum_{1}^{\infty} a_{n} \cos n x+\sum_{1}^{\infty} b_{n} \sin n x=\sum_{-\infty}^{\infty} c_{n} e^{i n x},\) use Euler's formula to find \(a_{n}\) and \(b_{n}\) in terms of \(c_{n}\) and \(c_{-n},\) and to find \(c_{n}\) and \(c_{-n}\) in terms of \(a_{n}\) and \(b_{n}\).

Short answer

Given \ f(x), \ a_{n} = c_{n} + c_{-n}, \ b_{n} = i (c_{n} - c_{-n}). \ c_{n} = \frac{a_{n}}{2} + \frac{ib_{n}}{2}, \ c_{-n} = \frac{a_{n}}{2} - \frac{ib_{n}}{2}.}

Step-by-step solution

- Recall Euler's Formula

Euler's formula states that for any real number \theta, \[ e^{i\theta} = \text{cos}(\theta) + i\text{sin}(\theta) \].Use this formula to express the cosine and sine functions in terms of exponentials.

- Rewrite f(x) Using Euler's Formula

Given\[ f(x) = \frac{1}{2} a_{0} + \text{sum}_{n=1}^{\text{∞}} a_{n} \text{cos}(nx) + \text{sum}_{n=1}^{\text{∞}} b_{n} \text{sin}(nx) \], rewrite \text{cos}(nx)\ and \text{sin}(nx)\ using Euler's formula:\[ \text{cos}(nx) = \frac{e^{inx} + e^{-inx}}{2}\] and \[ \text{sin}(nx) = \frac{e^{inx} - e^{-inx}}{2i}\].

- Substitute into the Series

Substitute the expressions for \text{cos}(nx)\ and \text{sin}(nx)\ into the original function:\[ f(x) = \frac{1}{2} a_{0} + \text{sum}_{n=1}^{\text{∞}} a_{n} \frac{e^{inx} + e^{-inx}}{2} + \text{sum}_{n=1}^{\text{∞}} b_{n} \frac{e^{inx} - e^{-inx}}{2i}\].

- Combine and Simplify the Series

Combine and simplify the series to match the exponential form \text{sum}_{n=-\text{∞}}^{\text{∞}} c_{n} e^{inx}:\[ f(x) = \frac{1}{2} a_{0} + \text{sum}_{n=1}^{\text{∞}} \frac{a_{n}}{2} (e^{inx} + e^{-inx}) + \text{sum}_{n=1}^{\text{∞}} \frac{b_{n}}{2i} (e^{inx} - e^{-inx})\].

- Equate Coefficients

Equate the coefficients of \text{each term} e^{inx} and e^{-inx}for the series \text{sum}_{n=-\text{∞}}^{\text{∞}} c_{n} e^{inx}. This gives us two equations:\[ c_{n} = \frac{a_{n}}{2} - \frac{b_{n}}{2i} = \frac{a_{n}}{2} + \frac{ib_{n}}{2} \] and \[ c_{-n} = \frac{a_{n}}{2} - \frac{b_{n}}{2i}\].

- Express Coefficients in Terms of Each Other

From the equations in step 5, solve for \ a_{n} \ and \ b_{n} \ in terms of \ c_{n} \ and \ c_{-n} \ :\[ a_{n} = c_{n} + c_{-n} \] and \[ b_{n} = i(c_{n} - c_{-n}) \].

- Final Expressions

Combine all the information to give:\[a_{n} = c_{n} + c_{-n}\] and \[ b_{n} = i (c_{n} - c_{-n})\]for \ c_{n} \ and \ c_{-n} \ we have:\[ c_{n} = \frac{a_{n}}{2} - \frac{b_{n}}{2i}\] and \[c_{-n}= \frac{a_{n}}{2} - \frac{b_{n}}{2i}\].

Key concepts

Euler's Formula

Euler's formula is a fundamental equation in complex analysis. It establishes a deeply insightful connection between complex exponentials and trigonometric functions. Euler's formula states that for any real number \theta\, the following relation holds:
\[ e^{i\theta} = \text{cos}(\theta) + i\text{sin}(\theta) \]
Here, \( e^{i\theta} \) represents a complex exponential, and the expression breaks down into a cosine term and an imaginary sine term.
This formula is especially useful because it allows us to convert trigonometric functions, like cosine and sine, into exponentials. For example:
\ e^{i nx} = \text{cos}(nx) + i \text{sin}(nx)\
and
\ e^{-i nx} = \text{cos}(nx) - i \text{sin}(nx)\

These relationships enable us to express trigonometric functions in exponential form, simplifying many calculations and derivations in the realm of signal processing and Fourier series.

Complex Exponentials

Complex exponentials provide a powerful way to handle periodic functions and signals in mathematics and engineering. They are especially useful in Fourier series, which decompose signals into their frequency components.
A complex exponential function is of the form \( e^{inx} \), where \ n \ is an integer, and \( x \) is a real variable.
Using Euler's formula, we can see that this function can be expressed as:
\[ e^{inx} = \text{cos}(nx) + i \text{sin}(nx) \]
Complex exponentials are crucial for transforming between time and frequency domains. For instance, in the Fourier series given in the problem, \( f(x) = \sum_{-\infty}^{\infty} c_n e^{inx} \), each term \( c_n e^{inx} \) represents a harmonic component of the signal.
Combining Euler’s formula with complex exponentials, we can convert between trigonometric and exponential forms of functions, which is essential in signal analysis.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in describing periodic phenomena. They appear frequently in engineering, physics, and signal processing.
In the context of Fourier series, these functions are paramount. A Fourier series represents periodic functions as sums of sine and cosine terms. For instance, we have
\( \text{cos}(nx) \) and \( \text{sin}(nx) \) in
\ f(x) = \frac{1}{2} a_{0} + \sum_{n=1}^{\infty} a_{n} \cos(nx) + b_{n} \sin(nx) \.
Using Euler’s formula, we can express these trigonometric functions in terms of exponentials:
\[ \text{cos}(nx) = \frac{e^{inx} + e^{-inx}}{2} \]
and
\[ \text{sin}(nx) = \frac{e^{inx} - e^{-inx}}{2i} \]
This conversion is crucial for simplifying computations and solving complex problems. In the given exercise, this transformation makes it easier to find relationships between different coefficients in the Fourier series, bridging the gap between the time and frequency domains.