Question

Find each of the following in the \(x+i y\) form and compare a computer solution. $$\sinh ^{-1}(i / \sqrt{2})$$

Short answer

The expression for \( \sinh^{-1}( \frac{i}{\sqrt{2}} ) \) in \( x + i y \) form is \( \frac{\log(2)}{2} + i \frac{\pi}{4} \).

Step-by-step solution

Understand the Given Problem

The exercise asks to find the expression for the inverse hyperbolic sine, \( \sinh^{-1} ( \frac{i}{\sqrt{2}} ) \), in the form \( x + i y \).

Use the Definition of Inverse Hyperbolic Sine

Recall that the inverse hyperbolic sine can be expressed as: \[ \sinh^{-1}(z) = \log(z + \sqrt{z^2 + 1}) \]

Substitute the Given Value

Substitute \( z = \frac{i}{\sqrt{2}} \) into the equation: \[ \sinh^{-1}( \frac{i}{\sqrt{2}} ) = \log \left( \frac{i}{\sqrt{2}} + \sqrt{ \left( \frac{i}{\sqrt{2}} \right)^2 + 1 } \right) \]

Simplify Inside the Square Root

Calculate \( \left( \frac{i}{\sqrt{2}} \right)^2 + 1 \): \[ \left( \frac{i}{\sqrt{2}} \right)^2 = \frac{i^2}{2} = \frac{-1}{2} \] \[ \left( \frac{-1}{2} + 1 \right) = \frac{1}{2} \] so it becomes: \[ \sqrt{ \frac{1}{2} } = \frac{1}{\sqrt{2}} \]

Combine and Simplify Further

Adding these terms together results in: \[ \frac{i}{\sqrt{2}} + \frac{1}{\sqrt{2}} = \frac{1 + i}{\sqrt{2}} \] so it becomes: \[ \sinh^{-1}(\frac{i}{\sqrt{2}}) = \log \left( \frac{1 + i}{\sqrt{2}} \right) \]

Simplify the Logarithmic Expression

Using properties of logarithms: \[ \log \left( \frac{1 + i}{\sqrt{2}} \right) = \log \left(\frac{1+i}{\sqrt{2}} \right) = \frac{-\log(2)}{2} + \log(1 + i) \]

Express in Terms of Natural Logarithms

The magnitude and angle of the complex number \( 1 + i \) are given as follows: \[ |1+i| = \sqrt{1^2 + 1^2} = \sqrt{2} \] \[ \theta = \tan^{-1} \left( \frac{1}{1} \right) = \frac{\pi}{4} \]

Final Logarithmic Form

Thus, we can express: \[ \log(1+i) = \log \sqrt{2} + i \frac{\pi}{4} = \frac{\log(2)}{2} + i \frac{\pi}{4} \]

Combine Results

Finally, add terms together to get the entire expression: \[ \sinh^{-1}(\frac{i}{\sqrt{2}}) = x + i y \]

Verify

Compare the results with a computer solution to verify the derived expression.

Key concepts

Inverse Hyperbolic Sine

Inverse hyperbolic functions provide a way to revert the hyperbolic sine, cosine, and tangent functions. In our specific problem, we are dealing with the inverse hyperbolic sine, commonly denoted as \(\text{sinh}^{-1}\). The mathematical expression of inverse hyperbolic sine is given by the formula: \[ \text{sinh}^{-1}(z) = \text{log}(z + \sqrt{z^2 + 1}) \]. This formula is essential because it transforms a hyperbolic sine function back to the original input value. Understanding and being able to manipulate this formula helps in finding solutions to various complex problems. When working with the inverse hyperbolic sine, remember:

• Hyperbolic functions are similar to trigonometric functions but are based on hyperbolas instead of circles.
• The inverse hyperbolic sine often involves logarithmic and square root operations.
• It’s crucial to understand how to simplify expressions and work with complex numbers to find the solution.

Complex Numbers

In the problem, we are given a complex number \(\frac{i}{\text{sqrt}(2)}\). Complex numbers are numbers that have both a real part and an imaginary part, with the imaginary unit denoted as \(i = \text{sqrt}(-1)\). Working with complex numbers typically involves:

• Addition, subtraction, multiplication, and division of complex numbers.
• Understanding properties of imaginary unit \(i\) like \(i^2 = -1\).
• Simplifying expressions that involve both real and imaginary parts.

In this exercise, we specifically performed the following steps:

• Calculate \( \text{sinh}^{-1}( \frac{i}{\text{sqrt}(2)} )\) which required us to substitute the complex number into the inverse hyperbolic function formula.
• Simplify the equation inside the logarithm by carefully handling the imaginary unit during squaring and adding operations.
• Finally, combine terms to present the result in the desired form \( x + i y \).

Logarithms

The next concept in our exercise involves logarithms. Logarithms are the inverses of exponential functions and are very useful in solving equations involving exponential growth or decay. In the context of the inverse hyperbolic sine function, logarithms helped simplify the formula: \[ \text{sinh}^{-1}(z) = \text{log}(z + \sqrt{z^2 + 1}) \]. Key properties of logarithms we used include:

• Logarithm of a product: \[ \text{log}(ab) = \text{log}(a) + \text{log}(b) \]
• Logarithm of a quotient: \[ \text{log}(a/b) = \text{log}(a) - \text{log}(b) \]
• Logarithm of a power: \[ \text{log}(a^b) = b \text{log}(a) \]

These properties were crucial when we simplified \[ \text{log}(\frac{1 + i}{\text{sqrt}(2)}) \]. By breaking it down, we were able to effectively manage and combine the steps in the solution. Understanding how logarithms work is vital in solving complex expressions and can aid significantly in various fields of mathematics and science.