Problem 5 - Advanced Engineering Mathematics

Question

Express \(e^{z}\) in the form \(a+i b\). \(z=\pi+\pi i\)

Step-by-Step Solution

Answer

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

Step 1: Understand Euler's Formula

Recall that Euler's formula states: \[e^{z} = e^{x+iy} = e^{x} ( ext{cos} \, y + i \, \text{sin} \, y)\]We are given \(z = \pi + \pi i\), which means \(x = \pi\) and \(y = \pi\). Our goal is to express the exponential of this complex number in terms of its real and imaginary components.

Step 2: Apply Euler's Formula

Using Euler's formula where \(x = \pi\) and \(y = \pi\), we substitute these into the formula:\[e^{z} = e^{\pi} (\text{cos} \, \pi + i \, \text{sin} \, \pi)\].

Step 3: Calculate Cosine and Sine Values

Calculate \(\text{cos} \, \pi\) and \(\text{sin} \, \pi\):- \(\text{cos} \, \pi = -1\)- \(\text{sin} \, \pi = 0\)

Step 4: Substitute Trigonometric Values

Now replace \(\text{cos} \, \pi\) and \(\text{sin} \, \pi\) back into the expression:\[e^{z} = e^{\pi} (-1 + i \, 0)\], which simplifies to \[-e^{\pi}\].

Step 5: Final Expression

Thus, the expression for \(e^{z}\) in the form \(a + ib\) is \(-e^{\pi} + i(0)\). This simplifies to just a real number:\(-e^{\pi}\).

Key Concepts

Euler's Formula

Euler's formula is the cornerstone of understanding complex exponential functions. Often written as \( e^{ix} = \cos x + i \sin x \), it bridges the gap between exponential and trigonometric functions.
This formula shows how a complex number with an imaginary exponent can be expressed in terms of cosine and sine functions. When dealing with a complex number in the form \( x + iy \), Euler's formula is adjusted to \( e^{x+iy} = e^{x}(\cos y + i \sin y) \).
This altered version allows us to separate the real exponential part, \( e^x \), from the oscillatory part defined by the trigonometric functions. This principle lays the foundation for complex analysis and is incredibly useful in various fields such as engineering, physics, and applied mathematics.

Complex Numbers

A complex number is a mathematical entity that extends the concept of one-dimensional real numbers to two dimensions by introducing an imaginary unit \( i \).
The standard form of a complex number is \( a + ib \), where \( a \) is the real part and \( b \) is the imaginary part. The imaginary unit \( i \) is defined as \( \sqrt{-1} \), making \( i^2 = -1 \).
These numbers are visualized on the complex plane, where the horizontal axis represents the real component and the vertical axis represents the imaginary component.

• Adding complex numbers is straightforward: \( (a+ib) + (c+id) = (a+c) + i(b+d) \).
• Multiplying them requires distribution: \( (a+ib)(c+id) = ac + aid + ibc + i^2bd \).

In solving problems like our original exercise, recognizing how to apply Euler's formula to complex numbers is crucial in simplifying and presenting results.

Trigonometric Functions

Trigonometric functions like cosine and sine are essential in Euler's formula.
The cosine function \( \cos \theta \) and sine function \( \sin \theta \) are fundamental for describing rotations, waves, and oscillations.
These functions are defined for any angle \( \theta \) and can take real or complex values.

• The cosine of \( \pi \) is \( -1 \), which reflects the position on the unit circle where the angle \( \pi \) lands on the negative x-axis.
• The sine of \( \pi \) is \( 0 \), indicating no vertical displacement at that angle.

When applying Euler's formula in our exercise, these specific values of sine and cosine at \( \pi \) simplify the complex expression, resulting in a real outcome of \( -e^{\pi} \). Understanding these trigonometric values helps interpret complex exponentials and their applications across various scientific and mathematical contexts.