Problem 72 - Thomas Calculus

Question

When \(a\) and \(b\) are real, we define \(e^{(a+i b) x}\) with the equation \begin{equation} e^{(a+i b) x}=e^{a x} \cdot e^{i b x}=e^{a x}(\cos b x+i \sin b x) \end{equation} Differentiate the right-hand side of this equation to show that \begin{equation} \frac{d}{d x} e^{(a+i b) x}=(a+i b) e^{(a+i b) x} \end{equation} Thus the familiar rule \((d / d x) e^{k x}=k e^{k x}\) holds for \(k\) complex as well as real.

Step-by-Step Solution

Answer

Differentiating \( e^{(a + i b)x} \) yields \((a + i b)e^{(a + i b)x}\), affirming the differentiation rule ( \(k\) complex or real).

Step 1: Differentiate the Exponential Function

Differentiate the expression \( e^{a x}(\cos b x + i \sin b x) \) using the product rule. Let \( u = e^{a x} \) and \( v = \cos b x + i \sin b x \). Using the product rule that states \( (uv)' = u'v + uv' \), we first find the derivatives of \( u \) and \( v \).

Step 2: Differentiate Each Function Separately

Find \( \frac{d}{dx} e^{a x} = a e^{a x} \) and \( \frac{d}{dx} (\cos b x + i \sin b x) = -b \sin b x + i b \cos b x \).

Step 3: Apply the Product Rule

Using the results from Step 2, apply the product rule: \( \frac{d}{d x}[e^{a x}(\cos b x + i \sin b x)] = (a e^{a x})(\cos b x + i \sin b x) + e^{a x}(-b \sin b x + i b \cos b x) \).

Step 4: Simplify the Expression

Rewrite the expression from Step 3 as \( a e^{a x}(\cos b x + i \sin b x) + e^{a x}(-b \sin b x + i b \cos b x) = e^{a x}[(a \cos b x - b \sin b x) + i(a \sin b x + b \cos b x)]\).

Step 5: Recognize the Complex Exponential Result

Notice that \( e^{a x}[(a \cos b x - b \sin b x) + i(a \sin b x + b \cos b x)] \) is equivalent to \((a + i b) e^{a x}(\cos b x + i \sin b x)\).

Step 6: Combine Terms to Show the Final Result

Combine terms from Step 5: \( (a + i b) e^{a x}(\cos b x + i \sin b x) = (a + i b) e^{(a + i b)x} \), confirming that \( \frac{d}{d x} e^{(a + i b)x} = (a + i b) e^{(a + i b)x} \).

Key Concepts

Exponential Functions

Exponential functions are critical in mathematics because they describe continuous growth or decay. They are expressed in the form of \( e^{kx} \), where \( e \) is Euler's number, approximately 2.718, and \( k \) is a constant. Here, \( k \) can be either a real or a complex number.

When dealing with complex exponentials like \( e^{(a+i b)x} \), we split them into their real and imaginary parts using Euler's formula. Euler's formula states that \( e^{i \theta} = \cos \theta + i \sin \theta \). This formula helps us represent complex exponentials in a form where both exponential growth (or decay) and oscillation are evident.

In the exercise, the expression \( e^{(a+i b)x} \) is split as \( e^{a x} (\cos b x + i \sin b x) \), combining both exponential functions and trigonometric functions. This versatile form makes it possible to differentiate complex exponentials using the tools we understand from real functions.

Product Rule in Calculus

The product rule is an essential technique in calculus for differentiating functions that are products of two other functions. If you have two functions \( u(x) \) and \( v(x) \), the product rule states that the derivative of their product \( u(x) \cdot v(x) \) is given by \( (uv)' = u'v + uv' \).

To apply the product rule to \( e^{a x} (\cos b x + i \sin b x) \), you first identify \( u(x) = e^{a x} \) and \( v(x) = \cos b x + i \sin b x \).

• Calculate \( u'(x) = a e^{a x} \).
• Calculate \( v'(x) = -b \sin b x + i b \cos b x \).
• Apply the product rule: \( \frac{d}{d x}[e^{a x} (\cos b x + i \sin b x)] = u'v + uv' \).

Using the derivatives calculated, you can simplify this expression to get the final differentiation result. This rule makes handling the complexity of multiplicative combinations of functions much more manageable.

Complex Numbers

Complex numbers extend the concept of real numbers to include an imaginary unit \( i \), where \( i^2 = -1 \). They are typically written in the form \( a + ib \), where \( a \) and \( b \) are real numbers.

In the context of the differentiation of complex exponentials, complex numbers allow us to blend real growth/decay (through \( e^{a x} \)) with oscillations (through the trigonometric components \( \cos b x + i \sin b x \)). This combination is at the heart of many physical phenomena, particularly in wave mechanics and electrical engineering.

Recognizing the interplay between the real and imaginary components helps us understand how these functions behave at a deeper level. When differentiating, complex numbers enable us to simplify expressions that would otherwise be quite unwieldy, streamlining the process and making the mathematics behind phenomena like alternating currents and waveforms more accessible.