Problem 18 - Complex Analysis

Question

Show that for \(\varphi \in \mathbb{R}-2 \pi \mathbb{Z}\) and for all \(n \in \mathbb{N}\) one has \(\begin{aligned} \frac{1}{2}+\sum_{\nu=1}^{n} \cos \nu \varphi &=\frac{\sin ((n+1 / 2) \varphi)}{2 \sin (\varphi / 2)} \\ \sum_{\nu=1}^{n} \sin \nu \varphi &=\frac{\sin (n \varphi / 2) \sin ((n+1) \varphi / 2)}{\sin (\varphi / 2)} \end{aligned}\)

Step-by-Step Solution

Answer

The formulas hold for the given conditions using Euler's formula and properties of geometric series.

Step 1: Recognize the Problem

We need to prove two trigonometric summation formulas involving the cosine and sine functions, valid for a specific range of \(\varphi\). The given condition \(\varphi \in \mathbb{R}-2\pi \mathbb{Z}\) implies that \(\varphi\) is not a multiple of \(2\pi\). Both identities depend on the integer \(n\).

Step 2: Apply Euler's Formula

Use Euler's formula \(e^{i\theta} = \cos \theta + i \sin \theta\) to express the sums in terms of exponential functions. Recognize that \(\sum_{u=1}^{n} \cos u \varphi\) can be combined using the formula for geometric series, leveraging Euler's identity.

Step 3: Derive Formula for Cosine Sum

We start by calculating \(\sum_{u=1}^{n} e^{i u \varphi}\): \[S_n = \sum_{u=1}^{n} e^{i u \varphi} = e^{i\varphi} + e^{2i\varphi} + \cdots + e^{ni\varphi}\] This is a geometric series with first term \(e^{i\varphi}\) and common ratio \(e^{i\varphi}\). Its sum can be expressed as \[S_n = \frac{e^{i\varphi}(e^{in\varphi} - 1)}{e^{i\varphi} - 1}\].

Step 4: Simplify Cosine Sum

The real part of \(S_n\): \[\sum_{u=1}^{n} \cos u \varphi = \Re(S_n) = \Re\left(\frac{e^{i\varphi}(e^{in\varphi} - 1)}{e^{i\varphi} - 1}\right)\] Simplify: \[\frac{1}{2} + \sum_{u=1}^{n} \cos u \varphi = \frac{\sin\left((n+\frac{1}{2})\varphi\right)}{2\sin(\frac{\varphi}{2})}\] using known exponential trigonometric identities.

Step 5: Derive Formula for Sine Sum

To find \(\sum_{u=1}^{n} \sin u \varphi\), we use the imaginary part of \(S_n\): \[\sum_{u=1}^{n} \sin u \varphi = \Im(S_n) = \Im\left(\frac{e^{i\varphi}(e^{in\varphi} - 1)}{e^{i\varphi} - 1}\right)\] Simplify using identities: \[\sum_{u=1}^{n} \sin u \varphi = \frac{\sin (\frac{n\varphi}{2}) \sin \left(\frac{(n+1)\varphi}{2}\right)}{\sin (\frac{\varphi}{2})}\].

Step 6: Verification of Results

Verify the derived expressions against the original formulas to ensure their correctness both algebraically and trigonometrically for the set \(\varphi \in \mathbb{R}-2\pi \mathbb{Z}\). Check through substitution of numbers and graphical representation if necessary.

Key Concepts

Euler's Formula

Euler's formula is a vital tool in understanding the relationship between trigonometric functions and exponential functions. It states that for any real number \( \theta \), the formula \( e^{i\theta} = \cos \theta + i \sin \theta \) holds. This means that complex exponentials can be split into their real and imaginary parts, namely the cosine and sine functions, respectively.

• Euler's formula helps to transform trigonometric problems into problems involving exponentials, which are often simpler to handle, especially in summations.
• By using Euler's formula, we can efficiently recognize and manipulate patterns in sums of trigonometric terms.

In our exercise, Euler's formula is used to convert the trigonometric terms \( \cos u \varphi \) and \( \sin u \varphi \) into exponential terms, facilitating the use of geometric series sums. This simplification utilizes the relation that the cosine and sine components form the real and imaginary parts of these exponential expressions, thereby encompassing them in a compact expression.

Geometric Series

The geometric series is a powerful mathematical concept that provides a way to sum sequences of terms that have a common ratio. For a geometric series with first term \( a \) and ratio \( r \), the sum of the first \( n \) terms is \( S_n = a \frac{r^n - 1}{r - 1} \), provided \( r eq 1 \).

• In the context of trigonometric sums, the sequence of complex exponentials forms a geometric series with \( e^{i\varphi} \) as the common ratio.
• This transforms the series \( e^{i\varphi} + e^{2i\varphi} + \cdots + e^{ni\varphi} \) into a solvable form using the geometric series sum formula.

Understanding and applying geometric series to complex trigonometric sums allows us to derive expressions for the sum of cosine or sine terms efficiently. For instance, in our exercise, the geometric series approach is pivotal in calculating sums involving \( \cos u \varphi \) and \( \sin u \varphi \) by first expressing them in exponential form and then applying the geometric series formula.

Trigonometric Identities

Trigonometric identities are mathematical equations involving trigonometric functions that are true for all values of the variable involved, where the functions are defined. They greatly simplify the process of proving more complex trigonometric statements.

• Identities such as \( \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \) are fundamental in breaking down and reconstructing expressions.
• The identities used in calculating sums of sine and cosine in our exercise blend these trigonometric identities with exponential identities obtained from Euler's formula.

By utilizing these identities, we can replace complex trigonometric expressions with simpler forms, making it easier to algebraically manipulate and prove the given formulas. In our case, expressing the sums of cosine and sine terms involved using trigonometric identities allows for a streamlined derivation of the results.

Cosine and Sine Functions

The cosine and sine functions are fundamental trigonometric functions that are used to describe wave patterns, oscillations, and circular motion. They are periodic functions defined for all real numbers and have important properties that make them suitable for various mathematical applications.

• Both functions are periodic with periods of \( 2\pi \), which means they repeat their values every \( 2\pi \) units.
• The cosine function \( \cos(\theta) \) measures the horizontal distance from the origin on the unit circle, while the sine function \( \sin(\theta) \) measures the vertical distance.

In trigonometric sums, these functions are often the focus of summations and transformations. In our exercise, the task involves summing series of cosine and sine terms, each dependent on an angle incrementally increasing by \( \varphi \). By manipulating these sums using their identities and properties, we achieve simplified forms, such as those showing sums of \( \cos u \varphi \) and \( \sin u \varphi \), pivotal in solving the trigonometric summation formulas at hand.