Problem 13 - Advanced Engineering Mathematics

Question

$$e^{-i z}=e^{y-x i}=e^{y} \cos x-i e^{y} \sin x$$

Step-by-Step Solution

Answer

The expression is already in its simplified form: \( e^{y-xi} = e^{y} \cos x - i e^{y} \sin x \).

Step 1: Analyze the Expression

The given expression is \( e^{-iz} = e^{y-xi} \). We need to interpret and simplify this expression using the properties of complex numbers and the exponential function.

Step 2: Apply Euler's Formula

Recall Euler's formula: \( e^{ix} = \cos x + i \sin x \). For the given expression \( e^{y-xi} \), consider this as \( e^y \cdot e^{-xi} \).

Step 3: Simplify the Exponential

Apply Euler's formula to \( e^{-xi} \):\[ e^{-xi} = \cos(-x) + i\sin(-x) = \cos x - i\sin x \]

Step 4: Combine the Terms

Multiply \( e^y \) by the result of Step 3:\[ e^{y-xi} = e^y (\cos x - i \sin x) \]

Step 5: Distribute and Simplify

Distribute \( e^y \) into the expression:\[ e^{y-xi} = e^y \cos x - i e^y \sin x \]

Step 6: Finalize the Expression

Observe that \( e^{y-xi} = e^y \cos x - i e^y \sin x \) matches the right side of the original equation. This confirms the equivalence.

Key Concepts

Euler's Formula

Euler's Formula is a beautiful bridge between exponential functions and trigonometry. It states that \( e^{ix} = \cos x + i \sin x \), where \( i \) is the imaginary unit. This formula is integral in simplifying expressions involving complex numbers. For example, to express \( e^{-ix} \), you can use the identities: \( \cos(-x) = \cos x \) and \( \sin(-x) = -\sin x \). Applying these to Euler's formula gives \( e^{-ix} = \cos x - i \sin x \). This highlights the symmetrical nature of sine and cosine with respect to negative angles, and provides a tool for manipulating complex exponential forms into trigonometric identities. Euler's Formula thus provides a fundamental understanding of how complex numbers interact within exponential functions, offering a unique perspective on the relationship between exponential growth and periodic motion.

Complex Numbers

Complex numbers are numbers of the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit, defined by \( i^2 = -1 \). They extend the concept of one-dimensional real numbers to the two-dimensional complex plane. Here, the horizontal axis represents real numbers, and the vertical axis represents imaginary numbers.
It's essential to understand that complex numbers can represent very different mathematical phenomena from real numbers.

• Magnitude of a complex number is given by \( |a+bi|=\sqrt{a^2+b^2} \).
• Complex conjugate of a number \( a + bi \) is \( a - bi \).
• Adding, subtracting, and multiplying complex numbers follow analogous rules as real numbers but include \( i \).

These properties make them useful in fields like electrical engineering and quantum physics where wave behaviors are prevalent. Understanding complex numbers enriches your understanding of a wide array of mathematical concepts.

Exponential Functions

Exponential functions are functions of the form \( f(x) = a^{bx} \), where \( a \) is a positive constant. They model a range of phenomena, including compound growth or decay, sound and light propagation, and continually compounded interest. One of the most significant exponential functions is \( e^x \), where \( e \approx 2.718 \).
In complex analysis, exponential functions take a new dimension. When complex numbers are involved, exponential functions can express oscillatory phenomena easily and effectively:

• They provide a connection between exponential and trigonometric functions through Euler's formula.
• They are critical in solving differential equations involving periodic components.
• They assist in describing waves and oscillations in terms of their amplitude and phase.

This flexibility makes exponential functions indispensable, particularly in engineering, physics, and mathematics itself, offering elegant solutions and descriptions of natural phenomena.