Problem 5 - Advanced Engineering Mathematics

Question

$$ \text { In Problems } 1-10 \text {, express } e^{z} \text { in the form } a+i b \text {. } $$ $$ z=\pi+\pi i $$

Step-by-Step Solution

Answer

The expression is \( -e^{\pi} + 0i \).

Step 1: Identify the Formula

To convert a complex exponential function into the form \( a + ib \), we use Euler's formula: \( e^{z} = e^{x+yi} = e^x (\cos y + i \sin y) \). Here, \( x \) and \( y \) are the real and imaginary parts of \( z \), respectively.

Step 2: Substitute the Values

Given \( z = \pi + \pi i \), identify \( x = \pi \) and \( y = \pi \). Substitute these into Euler's formula: \( e^{\pi + \pi i} = e^\pi (\cos \pi + i \sin \pi) \).

Step 3: Calculate the Trigonometric Functions

Calculate \( \cos \pi \) and \( \sin \pi \): \( \cos \pi = -1 \) and \( \sin \pi = 0 \).

Step 4: Substitute Trigonometric Values

Substitute the values from Step 3 into Euler's formula: \( e^{\pi}(\cos \pi + i \sin \pi) = e^{\pi}(-1 + 0i) \).

Step 5: Simplify the Expression

Simplify the expression: \( e^{\pi}(-1 + 0i) = -e^{\pi} + 0i \). This is already in the form \( a + ib \).

Step 6: Conclusion

The expression \( e^z = e^{\pi + \pi i} \) in the form \( a + ib \) is \( -e^{\pi} + 0i \), where \( a = -e^{\pi} \) and \( b = 0 \).

Key Concepts

Euler's Formula

One of the cornerstone principles in complex analysis is Euler's Formula. This remarkable equation bridges trigonometry and complex exponentials, connecting seemingly disparate branches of mathematics in an elegant way. Euler's formula is written as:\[ e^{i\theta} = \cos \theta + i \sin \theta\]

• Here, the variable \( \theta \) is the angle in radians.
• The formula represents the complex exponential \( e^{i\theta} \) in terms of sine and cosine functions.
• It is fundamental in converting exponential forms of complex numbers into trigonometric form, which is much easier to interpret and manipulate.

The beauty of Euler's formula lies not only in its simplicity but also in its applicability across various domains of mathematics and engineering. It is often used to simplify complex expressions and solve complex number equations. This formula is essential for understanding the exponential function of a complex number, as it allows us to break down and harness the functionalities of trigonometric functions simultaneously.

Complex Exponential Function

In mathematics, the complex exponential function extends the concept of exponentiation to complex numbers. For any complex number \( z = x + yi \) (where \( x \) and \( y \) are real numbers), we define the exponential function as:\[ e^z = e^{x + yi} = e^x \times e^{yi}\]

• The term \( e^x \) is the exponential of the real part of the complex number. It scales the magnitude of the resulting complex number.
• The term \( e^{yi} \) can be expressed using Euler’s formula as \( \cos y + i\sin y \).
• Thus, the complex exponential function is broken into real magnitude scaling and rotational transformation using trigonometric functions.

This formulation plays a crucial role in complex analysis and further fields such as electrical engineering. It expresses operations naturally in terms of oscillations, rotations, and growth represented by complex numbers. The complex exponential function is a powerful tool because it turns complex arithmetic into the manipulation of exponential magnitudes and trigonometric directions.

Trigonometric Functions

Trigonometric functions like sine and cosine are essential in the study of waves, oscillations, and circular motions. In complex analysis, these functions are vital in expressing the real and imaginary parts of complex numbers after exponential transformations.

• Cosine and sine in Euler's formula represent the real and imaginary axis projections of a unit circle in the complex plane.
• For example, \( \cos \pi = -1 \) and \( \sin \pi = 0 \) are values derived from their geometrical interpretations on the unit circle.
• In complex numbers, these functions explain the angular rotation, with \( \cos \theta\) giving the horizontal coordinate and \( i \sin \theta \) giving the vertical coordinate.

Understanding trigonometric functions is crucial for visualizing complex phenomena. In problems like our exercise, calculating \( \cos \) or \( \sin \) of given angles helps transform exponential expressions like \( e^{i\theta} \) into recognizable forms that represent rotations or directional growth.