Question

Find each of the following in the \(x+i y\) form and check your answers by computer. $$\sinh \left(1+\frac{i \pi}{2}\right)$$

Short answer

\[\sinh\left(1+\frac{i \pi}{2}\right) = i \frac{e + \frac{1}{e}}{2}\]

Step-by-step solution

- Understand the Hyperbolic Sine Function

The hyperbolic sine function is given by \[\sinh(z) = \frac{e^z - e^{-z}}{2}\]where \(z\) is a complex number. We need to use this definition to compute \(\sinh\left(1+\frac{i \pi}{2}\right)\).

- Express the Exponentials

Express \(z = 1 + \frac{i \pi}{2}\) and calculate the exponentials \(e^z\) and \(e^{-z}\). First, \(e^z = e^{1 + \frac{i \pi}{2}} = e \cdot e^{\frac{i \pi}{2}}\) Similarly, \(e^{-z} = e^{-(1 + \frac{i \pi}{2})} = e^{-1} e^{-\frac{i \pi}{2}}\).

- Use Euler's Formula

Using Euler's formula, \(e^{ix} = \cos(x) + i \sin(x)\), and substituting \(x = \frac{\pi}{2}\), we have: \(e^{\frac{i \pi}{2}} = \cos\left(\frac{\pi}{2}\right) + i \sin\left(\frac{\pi}{2}\right) = i\)Similarly, \(e^{-\frac{i \pi}{2}} = \cos\left(-\frac{\pi}{2}\right) + i \sin\left(-\frac{\pi}{2}\right) = -i\).

- Simplify the Expression

Now substitute these values back into the expressions for \(e^z\) and \(e^{-z}\):\(e^z = e \cdot i = i e\) \(e^{-z} = \frac{1}{e} \cdot (-i) = -\frac{i}{e}\).

- Compute \(\sinh(z)\)

Substitute into the original formula for \(\sinh(z)\): \[\sinh\left(1+\frac{i \pi}{2}\right) = \frac{i e - \left( -\frac{i}{e} \right)}{2}\] Simplify this to\[\sinh\left(1+\frac{i \pi}{2}\right) = \frac{i e + \frac{i}{e}}{2}\]

- Combine the Terms

Factor out the \(i\) and combine the terms: \[\sinh\left(1+\frac{i \pi}{2}\right) = \frac{i(e + \frac{1}{e})}{2}\]Which can be written as:\[\sinh\left(1+\frac{i \pi}{2}\right) = i \frac{e + \frac{1}{e}}{2}\]

- Verify the Result Using a Calculator

Verify the result by using a mathematical software or calculator to ensure accuracy. The expression \(\frac{e + \frac{1}{e}}{2}\) is the real part of the hyperbolic sine function evaluated at \(1\). So the complete result is correct if both parts coincide.

Key concepts

Complex Analysis

Complex analysis is a branch of mathematics that studies functions of complex numbers. A complex number has the form \(z = x + iy\) where \(x\) and \(y\) are real numbers, and \(i\) is the imaginary unit satisfying \(i^2 = -1\). Understanding complex numbers is crucial for calculating and simplifying functions involving them.

In the provided exercise, we deal with the complex number \(z = 1 + \frac{i \pi}{2}\). The hyperbolic sine function (\( \sinh(z) \)) used here is an extension of the sine function for complex-valued arguments. The function \(\sinh(z)\) itself is defined by:

• \[\sinh(z) = \frac{e^z - e^{-z}}{2}\]

This definition utilizes the exponential function which is core to many concepts in complex analysis. Calculating \(\sinh(z)\) implies working with these exponentials in the complex plane.

Euler's Formula

Euler's formula is a fundamental formula in complex analysis, relating complex exponentials to trigonometric functions. It states:

• \[e^{ix} = \cos(x) + i \sin(x)\]

In the solution, Euler's formula helps to express the complex exponentials \(e^{\frac{i \pi}{2}}\) and \(e^{-\frac{i \pi}{2}}\) in terms of sine and cosine functions.

According to Euler's formula:

• \[e^{\frac{i \pi}{2}} = \cos\left(\frac{\pi}{2}\right) + i \sin\left(\frac{\pi}{2}\right) = 0 + i = i\]
• And similarly, \[e^{-\frac{i \pi}{2}} = \cos\left(-\frac{\pi}{2}\right) + i \sin\left(-\frac{\pi}{2}\right) = 0 - i = -i\]

By substituting these back into the expressions for \(e^z\) and \(e^{-z}\), we simplify the original hyperbolic sine problem tremendously. This showcases the power of Euler's formula in simplifying calculations involving complex numbers.

Mathematical Computation

Mathematical computation is the process of calculating and simplifying expressions to find numerical or algebraic solutions. In this exercise, several steps involve performing precise calculations to simplify the original complex expression \(\sinh\left(1+\frac{i \pi}{2}\right)\).

• First, we express the exponentials \(e^z\) and \(e^{-z}\).
• Next, we use previously derived expressions from Euler's formula: \[e^z = e \cdot i = ie\]
• Similarly, \[e^{-z} = \frac{1}{e} \cdot (-i) = -\frac{i}{e}\]

We then substitute these back into the formula for \(\sinh(z)\):

• \[\sinh\left(1+\frac{i \pi}{2}\right) = \frac{ie - \left( -\frac{i}{e} \right)}{2} = \frac{ie + \frac{i}{e}}{2}\]
• By factoring out \(i\) and combining terms, we get:
• \[\sinh\left(1+\frac{i \pi}{2}\right) = \frac{i(e + \frac{1}{e})}{2} = i \frac{e + \frac{1}{e}}{2}\]

This final result can be confirmed by using a calculator or mathematical software to verify its accuracy. This process highlights the careful steps and computational techniques used in the realm of complex analysis.