Problem 36
Question
In Problems 33 and 34, approximate the value of the given expression using the indicated number of terms of a Maclaurin series. Use the Maclaurin series for \(e^{i z}\) to prove Euler's formula for complex \(z\) : $$ e^{i z}=\cos z+i \sin z $$
Step-by-Step Solution
Verified Answer
Euler's formula is proven by substituting \(iz\) into the exponential Maclaurin series and separating real and imaginary components.
1Step 1: Recall the Maclaurin Series for Exponential Functions
The Maclaurin series for the exponential function \(e^x\) is given by:\[e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \]This series expansion holds for any real or complex \(x\).
2Step 2: Substitute in the Complex Argument
Replace \(x\) with \(iz\) in the series to get the Maclaurin series for \(e^{iz}\):\[e^{iz} = 1 + iz + \frac{(iz)^2}{2!} + \frac{(iz)^3}{3!} + \frac{(iz)^4}{4!} + \cdots \]
3Step 3: Simplify Complex Powers
Calculate the powers of \(iz\):\( (iz)^2 = i^2z^2 = -z^2 \)\( (iz)^3 = i^3z^3 = -iz^3 \)\( (iz)^4 = i^4z^4 = z^4 \)Thus, the series simplifies to:\[e^{iz} = 1 + iz - \frac{z^2}{2!} - \frac{iz^3}{3!} + \frac{z^4}{4!} + \cdots \]
4Step 4: Separate Real and Imaginary Components
Group terms to separate the real and imaginary parts:The real part: \[1 - \frac{z^2}{2!} + \frac{z^4}{4!} + \cdots = \cos z\]The imaginary part: \[iz - \frac{iz^3}{3!} + \frac{iz^5}{5!} + \cdots = i\sin z\]This gives us the expression:\[e^{iz} = \cos z + i\sin z\]
5Step 5: Conclude using Euler's Formula
From the above derivation, the Maclaurin series for \(e^{iz}\) corresponds precisely to the sum of \(\cos z\) and \(i\sin z\), thus proving Euler's formula:\[e^{iz} = \cos z + i\sin z\]
Key Concepts
Maclaurin SeriesComplex AnalysisExponential Functions
Maclaurin Series
The Maclaurin series is an important tool in calculus, especially when dealing with functions like exponential functions. This series represents a function as an infinite sum of terms calculated from its derivatives at a single point, usually zero. For the exponential function \(e^x\), its Maclaurin series is \[e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \]. Here, \(x\) can be a real or a complex number, which gives this series a wide range of applications.
When using this series to work with complex numbers, like substituting \(x\) with \(iz\) where \(i\) is the imaginary unit, you can explore interesting relationships such as Euler's formula. Euler's formula elegantly links complex exponentials and trigonometry and shows how \(e^{iz}\) can be expressed as a combination of cosine and sine functions.
When using this series to work with complex numbers, like substituting \(x\) with \(iz\) where \(i\) is the imaginary unit, you can explore interesting relationships such as Euler's formula. Euler's formula elegantly links complex exponentials and trigonometry and shows how \(e^{iz}\) can be expressed as a combination of cosine and sine functions.
- This demonstrates the versatility and power of the Maclaurin series in both real and complex analysis contexts.
- It helps to simplify and understand complex exponentials in terms of familiar trigonometric sine and cosine functions.
Complex Analysis
Complex analysis is the study of complex numbers and functions involving these numbers. A foundational concept in this field is Euler's formula, which states \(e^{iz} = \cos z + i \sin z\). This fascinating formula forms a bridge between complex exponentials and trigonometric functions.
To derive Euler's formula using the Maclaurin series, one substitutes a complex argument \(iz\) into the exponential function's series. By simplifying the powers of \(iz\), resulting in a mixture of real and imaginary components, these can be grouped into separate parts representing the series for cosine and sine. The formula then allows for an insightful representation of complex numbers in the form of \(e^{ix}\), linking exponential growth and oscillation.
To derive Euler's formula using the Maclaurin series, one substitutes a complex argument \(iz\) into the exponential function's series. By simplifying the powers of \(iz\), resulting in a mixture of real and imaginary components, these can be grouped into separate parts representing the series for cosine and sine. The formula then allows for an insightful representation of complex numbers in the form of \(e^{ix}\), linking exponential growth and oscillation.
- Complex analysis reveals hidden beauty and interconnectedness within mathematical concepts.
- This also enables the solveability of complex integrals and differential equations.
Exponential Functions
Exponential functions are a category of mathematical functions that show constant growth rates. The general form is \(e^x\), where \(e\) is Euler's number, approximately equal to 2.71828. One particular challenge comes when dealing with exponentials of complex numbers, like in Euler's formula.
The connection between exponential functions and complex numbers is highlighted through the Maclaurin series expansion of \(e^{iz}\). The exponential function \(e^{iz}\) overlaps with trigonometric functions, letting us express it as \( \cos z + i \sin z\). This connection is not just academic; it has practical applications in fields like engineering and physics, where wave phenomena and alternating circuits can be analyzed.
The connection between exponential functions and complex numbers is highlighted through the Maclaurin series expansion of \(e^{iz}\). The exponential function \(e^{iz}\) overlaps with trigonometric functions, letting us express it as \( \cos z + i \sin z\). This connection is not just academic; it has practical applications in fields like engineering and physics, where wave phenomena and alternating circuits can be analyzed.
- Exponential functions with complex inputs describe oscillating systems incredibly effectively.
- This makes them essential for modeling sophisticated modern technologies.
Other exercises in this chapter
Problem 36
The steady-state temperature \(u(x, y)\) in a semi-infinite plate is determined from $$\begin{aligned}&\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2}
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Approximate the value of the given expression using the indicated number of terms of a Maclaurin series. Use the Maclaurin series for \(e^{i z}\) to prove Euler
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In Section \(15.1\) we defined the error function as $$ \operatorname{erf}(z)=\frac{2}{\sqrt{\pi}} \int_{0}^{z} e^{-t^{2}} d t $$ Find a Maclaurin series for \(
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