Chapter 2: Problem 4
Find one or more values of each of the following complex expressions and compare with a computer solution. $$\sinh (1+i \pi / 2)$$
Short Answer
Expert verified
The value is \( \frac{i(e + \frac{1}{e})}{2} \).
Step by step solution
01
Understand the definition of hyperbolic sine
The hyperbolic sine function is defined as \(\text{sinh}(z) = \frac{e^z - e^{-z}}{2}\), where \(z\) is a complex number.
02
Substitute the given complex number into the definition
For \(z = 1 + i \frac{\pi}{2}\), substitute into the hyperbolic sine definition: \(\text{sinh}(1 + i \frac{\pi}{2}) = \frac{e^{1 + i \frac{\pi}{2}} - e^{-(1 + i \frac{\pi}{2})}}{2}\).
03
Simplify the exponentials using Euler's formula
Using Euler's formula \(e^{ix} = \cos(x) + i \sin(x)\), we find: \(e^{1 + i \frac{\pi}{2}} = e^1 e^{i \frac{\pi}{2}} = e \left( \cos(\frac{\pi}{2}) + i \sin(\frac{\pi}{2}) \right) = e(i) = ei\) and \(e^{-(1 + i \frac{\pi}{2})} = \frac{1}{e^{1 + i \frac{\pi}{2}}} = \frac{1}{ei} = -\frac{i}{e}\)
04
Combine the terms
Combine the terms: \(\text{sinh}(1 + i \frac{\pi}{2}) = \frac{ei - (-\frac{i}{e})}{2} = \frac{ei + \frac{i}{e}}{2} \).
05
Simplify the expression
To simplify: \(\text{sinh}(1 + i \frac{\pi}{2}) = \frac{i(e + \frac{1}{e})}{2} \). Since \(e = 2.718\), approximate or leave the expression in this form.
06
Compare with a computer solution
Using a computer solution (e.g., a calculator or mathematical software like WolframAlpha or MATLAB), the value can be verified. The computer can give a more precise decimal value to compare with the symbolic answer.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
hyperbolic functions
Hyperbolic functions, similar to trigonometric functions, are defined using exponential functions. They often appear in complex analysis. The hyperbolic sine function, denoted as \text{sinh}(z)\, is defined specifically by the formula:
\text{sinh}(z) = \frac{e^z - e^{-z}}{2}\.
In this exercise, we deal with the complex number \(z = 1 + i \frac{\pi}{2}\). This makes it crucial to grasp the behavior of hyperbolic functions with complex inputs. Hyperbolic functions can be particularly useful because they satisfy similar identities to trigonometric functions but are applied in different contexts, such as theories involving hyperbolas and in different areas of complex analysis.
\text{sinh}(z) = \frac{e^z - e^{-z}}{2}\.
In this exercise, we deal with the complex number \(z = 1 + i \frac{\pi}{2}\). This makes it crucial to grasp the behavior of hyperbolic functions with complex inputs. Hyperbolic functions can be particularly useful because they satisfy similar identities to trigonometric functions but are applied in different contexts, such as theories involving hyperbolas and in different areas of complex analysis.
Euler's formula
Euler's formula establishes a fundamental relationship between exponential functions and trigonometric functions. It states that
\( e^{ix} = \cos(x) + i \sin(x) \).
This is exceptionally useful when dealing with complex exponentials. In the given exercise, you need to utilize Euler's formula to simplify the expression involving the hyperbolic sine function. For instance,
\(e^{1 + i \frac{\pi}{2}} = e^1 e^{i \frac{\pi}{2}} = e \left( \cos(\frac{\pi}{2}) + i \sin(\frac{\pi}{2}) \right) = e(i) = ei\).
Similarly, the negative exponent can be simplified using:
\(e^{-(1 + i \frac{\pi}{2})} = \frac{1}{e^{1 + i \frac{\pi}{2}}}\ = \frac{1}{ei} = -\frac{i}{e}\).
Understanding Euler's formula thus provides the gateway to simplifying and interpreting expressions in complex analysis.
\( e^{ix} = \cos(x) + i \sin(x) \).
This is exceptionally useful when dealing with complex exponentials. In the given exercise, you need to utilize Euler's formula to simplify the expression involving the hyperbolic sine function. For instance,
\(e^{1 + i \frac{\pi}{2}} = e^1 e^{i \frac{\pi}{2}} = e \left( \cos(\frac{\pi}{2}) + i \sin(\frac{\pi}{2}) \right) = e(i) = ei\).
Similarly, the negative exponent can be simplified using:
\(e^{-(1 + i \frac{\pi}{2})} = \frac{1}{e^{1 + i \frac{\pi}{2}}}\ = \frac{1}{ei} = -\frac{i}{e}\).
Understanding Euler's formula thus provides the gateway to simplifying and interpreting expressions in complex analysis.
exponential functions
Exponential functions are key in both real and complex analysis. For a complex number \(z\), the exponential function is given by
\( e^z = e^{x + iy} = e^x \cdot e^{iy} \), where \(x\) and \(y\) are real numbers.
Using Euler's formula, \( e^{iy} \) translates to \( \cos(y) + i \sin(y) \). This allows exponential functions to be represented in a way that involves sines and cosines.
Therefore, in the context of the exercise, the expressions \( e^{1 + i \frac{\pi}{2}} \) and \( e^{-(1 + i \frac{\pi}{2})} \) involve calculating these exponential terms and express them in a simpler trigonometric form. This further leads to solving or simplifying complex expressions efficiently.
\( e^z = e^{x + iy} = e^x \cdot e^{iy} \), where \(x\) and \(y\) are real numbers.
Using Euler's formula, \( e^{iy} \) translates to \( \cos(y) + i \sin(y) \). This allows exponential functions to be represented in a way that involves sines and cosines.
Therefore, in the context of the exercise, the expressions \( e^{1 + i \frac{\pi}{2}} \) and \( e^{-(1 + i \frac{\pi}{2})} \) involve calculating these exponential terms and express them in a simpler trigonometric form. This further leads to solving or simplifying complex expressions efficiently.
complex numbers
Complex numbers combine real and imaginary components and are expressed in the form
\( z = a + bi \), where \(a\) and \(b\) are real numbers, and \( i\) is the imaginary unit defined by \(i^2 = -1\).
They expand the one-dimensional number line to a two-dimensional complex plane. This allows for the unification of algebraic properties with geometric interpretations.
When using complex numbers in hyperbolic functions and exponential expressions, it's essential to handle both parts of the numbers carefully. In the given exercise, the complex number \(1 + i \frac{\pi}{2}\) enters the hyperbolic sine function. Therefore, working with both the real and imaginary parts throughout the calculations becomes necessary for accurate results. The understanding of complex numbers thus forms the basis for exploring more intricate functions and relationships in complex analysis.
\( z = a + bi \), where \(a\) and \(b\) are real numbers, and \( i\) is the imaginary unit defined by \(i^2 = -1\).
They expand the one-dimensional number line to a two-dimensional complex plane. This allows for the unification of algebraic properties with geometric interpretations.
When using complex numbers in hyperbolic functions and exponential expressions, it's essential to handle both parts of the numbers carefully. In the given exercise, the complex number \(1 + i \frac{\pi}{2}\) enters the hyperbolic sine function. Therefore, working with both the real and imaginary parts throughout the calculations becomes necessary for accurate results. The understanding of complex numbers thus forms the basis for exploring more intricate functions and relationships in complex analysis.
