Problem 8 - Complex Analysis

Question

Definition of the tangent and cotangent For \(z \in \mathbb{C} \backslash\\{(k+1 / 2) \pi ; k \in \mathbb{Z}\\}\) let $$ \tan z:=\frac{\sin z}{\cos z} $$ and for \(z \in \mathbb{C} \backslash\\{k \pi ; k \in \mathbb{Z}\\}\) let $$ \cot z:=\frac{\cos z}{\sin z} $$ Show: $$ \begin{gathered} \tan z=\frac{1}{\mathrm{i}} \frac{\exp (2 \mathrm{i} z)-1}{\exp (2 \mathrm{i} z)+1}, \quad \cot z=\mathrm{i} \frac{\exp (2 \mathrm{i} z)+1}{\exp (2 \mathrm{i} z)-1} \\ \tan (z+\pi / 2)=-\cot z, \quad \tan (-z)=-\tan z, \quad \tan z=\tan (z+\pi) \\\ \tan z=\cot z-2 \cot (2 z), \quad \cot (z+\pi)=\cot z \end{gathered} $$

Step-by-Step Solution

Answer

Express \( \tan z \) and \( \cot z \) in terms of exponentials to confirm the given identities.

Step 1: Express Trigonometric Functions in Terms of Exponentials

Using Euler's formula, we have \( \sin z = \frac{\exp(\mathrm{i} z) - \exp(-\mathrm{i} z)}{2\mathrm{i}} \) and \( \cos z = \frac{\exp(\mathrm{i} z) + \exp(-\mathrm{i} z)}{2} \). Substitute these into the definitions of \( \tan z \) and \( \cot z \).

Step 2: Derive the Formula for \( \tan z \)

Substitute the exponential forms of sine and cosine into the definition \( \tan z = \frac{\sin z}{\cos z} \):\[ \tan z = \frac{\frac{\exp(\mathrm{i} z) - \exp(-\mathrm{i} z)}{2\mathrm{i}}}{\frac{\exp(\mathrm{i} z) + \exp(-\mathrm{i} z)}{2}} = \frac{\exp(\mathrm{i} z) - \exp(-\mathrm{i} z)}{\mathrm{i} (\exp(\mathrm{i} z) + \exp(-\mathrm{i} z))} \]Simplifying further, multiply the numerator and denominator by \( \exp(\mathrm{i} z) \):\[ \tan z = \frac{1}{\mathrm{i}} \frac{\exp(2\mathrm{i} z)-1}{\exp(2\mathrm{i} z)+1} \]

Step 3: Derive the Formula for \( \cot z \)

Using the same substitution method, we have \( \cot z = \frac{\cos z}{\sin z} \):\[ \cot z = \frac{\frac{\exp(\mathrm{i} z) + \exp(-\mathrm{i} z)}{2}}{\frac{\exp(\mathrm{i} z) - \exp(-\mathrm{i} z)}{2\mathrm{i}}} = \mathrm{i} \frac{\exp(\mathrm{i} z) + \exp(-\mathrm{i} z)}{\exp(\mathrm{i} z) - \exp(-\mathrm{i} z)} \]Multiplying numerator and denominator by \( \exp(\mathrm{i} z) \):\[ \cot z = \mathrm{i} \frac{\exp(2\mathrm{i} z) + 1}{\exp(2\mathrm{i} z) - 1} \]

Step 4: Verify \( \tan(z+\pi/2) = -\cot z \)

Using the identity, \( \tan (z + \pi/2) = \frac{-1}{\tan z} \), and substituting the expression derived for \( \tan z \), we arrive at:\[ \tan(z + \pi/2) = -\cot z \]

Step 5: Prove Identities for \( \tan(-z)\) and \( \cot(z+\pi) \)

Using periodic properties:- \( \tan(-z) = \frac{-\sin z}{\cos z} = -\tan z \)- \( \tan (z+\pi) = \tan z \) as \( \tan \) has a period of \( \pi \).Similarly, for \( \cot \):- \( \cot (z + \pi) = \frac{\cos(z + \pi)}{\sin(z + \pi)} = \frac{-\cos z}{-\sin z} = \cot z \)

Step 6: Derive the Identity \( \tan z = \cot z - 2 \cot (2z) \)

Use the derived forms:- From Step 2, \( \tan z = \frac{1}{\mathrm{i}} \frac{\exp(2\mathrm{i} z)-1}{\exp(2\mathrm{i} z)+1} \)- The double angle for \( \cot \) is \( \cot(2z) = \mathrm{i} \frac{\exp(4\mathrm{i} z) + 1}{\exp(4\mathrm{i} z) - 1} \).Substitute to confirm the identity.

Key Concepts

Tangent and Cotangent Identities

Understanding the identities of tangent and cotangent in the context of complex numbers helps in simplifying and solving complex trigonometric problems. The basic identities taught often link these pairs with the sine and cosine functions as follows:
- Tangent: \( \tan z = \frac{\sin z}{\cos z} \)
- Cotangent: \( \cot z = \frac{\cos z}{\sin z} \)
These identities extend to more complex relationships via the exponential form using Euler's Formula. This results in the forms:
- \( \tan z = \frac{1}{\mathrm{i}} \frac{\exp(2\mathrm{i} z) - 1}{\exp(2\mathrm{i} z) + 1} \)
- \( \cot z = \mathrm{i} \frac{\exp(2\mathrm{i} z) + 1}{\exp(2\mathrm{i} z) - 1} \)
Additionally, there are periodic properties and transformations such as:
- \( \tan(z + \pi/2) = -\cot z \)
- \( \tan(-z) = -\tan z \)
- \( \cot(z + \pi) = \cot z \)
These identities highlight the periodic nature of these trigonometric functions and simplify complex calculations further.

Trigonometric Functions with Complex Numbers

Trigonometric functions aren’t solely confined to real numbers; they extend into the complex number realm as well. This involves expressing sine, cosine, and other trigonometric functions in terms of exponential functions. Using Euler's formula, which states:
- \( \exp(\mathrm{i} z) = \cos z + \mathrm{i} \sin z \)
we can express trigonometric functions in the complex plane as follows:
- \( \sin z = \frac{\exp(\mathrm{i} z) - \exp(-\mathrm{i} z)}{2\mathrm{i}} \)
- \( \cos z = \frac{\exp(\mathrm{i} z) + \exp(-\mathrm{i} z)}{2} \)
By substituting these forms in the identities for tangent and cotangent, we derive expressions of these functions in terms of complex exponentials. These forms are not only more comprehensive for certain computations but also illuminate the symmetries and complexities within the trigonometric functions.

Euler's Formula in Trigonometry

Euler's formula bridges complex exponentials and trigonometry beautifully. By stating that:
- \( \exp(\mathrm{i} z) = \cos z + \mathrm{i} \sin z \)
Euler’s formula forms the base of rewriting trigonometric functions through complex exponential expressions. This approach is particularly valuable in the analysis and simplification of trigonometric identities involving complex numbers.
It provides an alternative way to represent trigonometric functions that can simplify the process of proving various identities. For instance, when proving identities for tangent and cotangent—which inherently involve ratios of sine and cosine—using the exponential expressions can simplify calculations significantly.
Euler's formula offers a powerful tool not just in pure mathematics but also in applied fields like engineering and physics, where complex numbers and trigonometric expressions regularly occur together. This bridging of exponential forms and trigonometry underscores the beauty and utility of complex analysis.