Problem 10 - Advanced Engineering Mathematics

Question

Express \(e^{z}\) in the form \(a+i b\). \(z=-0.23-i\)

Step-by-Step Solution

Answer

Expressing the complex exponential \( e^{z} \) gives \( 0.429 - 0.668i \).

Step 1: Identify the Complex Number

The complex number is given as \( z = -0.23 - i \). Here, \( z \) can be expressed in the form \( z = x + yi \) where \( x = -0.23 \) and \( y = -1 \).

Step 2: Exponential Form of a Complex Number

The exponential form of a complex number \( e^{z} \) can be expanded using Euler's formula as \( e^{x+yi} = e^{x}(\cos(y) + i\sin(y)) \).

Step 3: Apply Euler's Formula

Substitute \( x = -0.23 \) and \( y = -1 \) into the Euler's formula: \[ e^{z} = e^{-0.23}(\cos(-1) + i\sin(-1)) \].

Step 4: Calculate the Exponential Term

Calculate \( e^{-0.23} \). This value is approximately \( 0.7942 \).

Step 5: Calculate the Trigonometric Values

Find \( \cos(-1) \) and \( \sin(-1) \) using a calculator. They are approximately \( \cos(-1) \approx 0.5403 \) and \( \sin(-1) \approx -0.8415\).

Step 6: Multiply the Terms

Multiply \( e^{-0.23} \) by the results from the trigonometric functions: \[ e^{z} = 0.7942 (0.5403 + i(-0.8415)) \].

Step 7: Expand the Equation

Expand the equation: \[ e^{z} = 0.7942 \times 0.5403 + 0.7942 \times (-0.8415) i \].

Step 8: Final Calculation

Perform the multiplications: \[ 0.7942 \times 0.5403 = 0.429 \]\[ 0.7942 \times (-0.8415) = -0.668 \]Thus, \( e^{z} = 0.429 - 0.668i \).

Key Concepts

Euler's Formula

Euler's Formula provides a profound connection between complex numbers and trigonometry. It's a cornerstone in understanding complex exponential functions. In simple terms, Euler's Formula states: \[ e^{ix} = \cos(x) + i\sin(x) \] Here, \( e \) is the base of the natural logarithm, \( i \) is the imaginary unit, and \( x \) is a real number. This formula beautifully encapsulates how the exponential function can describe rotation in the complex plane.

• **Exponential Form**: Any complex number \( z = x + yi \), where \( x \) and \( y \) are real numbers, can be expressed as \( e^{z} = e^{x+yi} \).
• **Application**: Euler's Formula helps convert complex numbers from polar to rectangular forms, which facilitates easier calculations.

For example, in the original problem, Euler's Formula helps transform the expression \( e^{z} \) into \( e^{-0.23}(\cos(-1) + i\sin(-1)) \). This way, it separates the real and imaginary parts, which makes calculation straightforward.

Complex Numbers

Complex numbers are an extension of the familiar real numbers and encompass both a real and an imaginary component. A complex number takes the form \( z = a + bi \), where \( a \) is the real part and \( b \) is the imaginary part.

• **Imaginary Unit \( i \)**: Represents the square root of \(-1\). Since no real number satisfies this, it's an essential component of complex numbers.
• **Visual Representation**: These numbers can be visualized on the complex plane, where the horizontal axis represents real numbers and the vertical axis represents imaginary numbers.
• **Arithmetic Operations**: Addition and subtraction of complex numbers occur component-wise, and multiplication involves distributing terms while using \( i^2 = -1 \).

In our exercise, the complex number \( z = -0.23 - i \) features a real part of \(-0.23\) and an imaginary part of \(-1\). Understanding this allows for the use of Euler's Formula to express the exponential form as a complex number with both real and imaginary parts.

Trigonometric Functions

Trigonometric functions like sine and cosine play a crucial role in expressing complex exponential functions using Euler's Formula. When dealing with complex numbers in exponential form, these functions help describe the rotation and scaling on the complex plane.

• **Cosine Function (\( \cos(x) \))**: Provides the horizontal distance from the origin on the unit circle. It's the real component when using Euler’s Formula.
• **Sine Function (\( \sin(x) \))**: Offers the vertical distance from the origin on the unit circle and accounts for the imaginary component.
• **Key Angle Calculations**: These values can be computed using calculators for non-standard angles like \(-1\), often giving results up to four decimal places: \( \cos(-1) \approx 0.5403 \), \( \sin(-1) \approx -0.8415 \).

In our solution, these functions are pivotal for calculating \( e^{z} \) by determining the contributions of both trigonometric components: \( e^{-0.23}(0.5403 + i(-0.8415)) \), which helps expand and finalizes the calculation in the problem.