Question

Find one or more values of each of the following complex expressions in the easiest way you can. \(\sinh (1+i \pi / 2)\)

Short answer

\( \sinh(1 + i \frac{\pi}{2}) = \frac{i(e + \frac{1}{e})}{2} \)

Step-by-step solution

Understand the Hyperbolic Sine Function

The hyperbolic sine function is defined as \[ \sinh(z) = \frac{e^z - e^{-z}}{2} \] where \(z\) is a complex number.

Substitute The Complex Number

Substitute \(z = 1 + i \frac{\pi}{2}\) into the hyperbolic sine function. Thus, \[ \sinh(1 + i \frac{\pi}{2}) = \frac{e^{1 + i \frac{\pi}{2}} - e^{-(1 + i \frac{\pi}{2})}}{2} \]

Simplify The Exponentials

Use Euler's formula which states that \( e^{ix} = \cos(x) + i \sin(x) \) to simplify each exponential term. Compute separately: \[ e^{1 + i \frac{\pi}{2}} = e^1 \cdot e^{i \frac{\pi}{2}} = e \cdot (\cos(\frac{\pi}{2}) + i \sin(\frac{\pi}{2})) = e \cdot (0 + i \cdot 1) = ie \] Similarly, \[ e^{-(1 + i \frac{\pi}{2})} = e^{-1} \cdot e^{-i \frac{\pi}{2}} = \frac{1}{e} \cdot (\cos(- \frac{\pi}{2}) + i \sin(- \frac{\pi}{2})) = \frac{1}{e} \cdot (0 - i \cdot 1) = - \frac{i}{e} \]

Combine and Simplify

Now combine the simplified exponential terms into the hyperbolic sine formula: \[ \sinh(1 + i \frac{\pi}{2}) = \frac{ie - (- \frac{i}{e})}{2} = \frac{ie + \frac{i}{e}}{2} = \frac{i(e + \frac{1}{e})}{2} \]

Key concepts

hyperbolic sine

The hyperbolic sine function, denoted as \(\text{sinh}(z)\), is analogous to the sine function but for hyperbolic geometry instead of circular geometry. It is defined as: \[\text{sinh}(z) = \frac{e^z - e^{-z}}{2}\] where \(z\) is a complex number. This function helps in solving problems involving complex numbers and exponential functions in hyperbolic contexts. It differs from the ordinary sine function but shares similar properties such as being odd (i.e., \(\text{sinh}(-z) = -\text{sinh}(z)\)). The hyperbolic sine function arises naturally in various fields including engineering, physics, and mathematical finance due to its relevance in growth and decay processes.

complex numbers

Complex numbers are numbers of the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit satisfying \(i^2 = -1\). These numbers extend the real numbers and are essential in solving equations that cannot be solved using only real numbers. In our exercise, \(1 + i \frac{\text{π}}{2}\) is a complex number. Complex numbers can be represented graphically using the complex plane, where the horizontal axis represents the real part and the vertical axis represents the imaginary part. They are useful in various applications including electrical engineering and fluid dynamics, where they provide a natural way to describe oscillations and waves.

Euler's formula

Euler's formula is a fundamental equation in complex analysis given by \(e^{ix} = \text{cos}(x) + i\text{sin}(x)\). This elegant formula links exponential functions with trigonometric functions, providing deep insights into complex number theory. In the given exercise, we apply Euler's formula to simplify the terms \(e^{i\frac{\text{π}}{2}}\) and \(e^{-i\frac{\text{π}}{2}}\). Specifically, \(e^{i \frac{\text{π}}{2}} = \text{cos}(\frac{\text{π}}{2}) + i\text{sin}(\frac{\text{π}}{2}) = 0 + i = i\), and similarly, \(e^{-i \frac{\text{π}}{2}} = \text{cos}(-\frac{\text{π}}{2}) + i\text{sin}(-\frac{\text{π}}{2}) = 0 - i = -i\). These transformations simplify the complex exponential expressions and are combined further on.

exponential function

The exponential function, denoted as \(e^z\) where \(e\) is the base of natural logarithms (~2.718), is a crucial function in mathematics. It grows or decays exponentially and appears in various contexts like compound interest, population growth, and radioactive decay. When extended to complex numbers, the exponential function still retains similar properties but also gains fascinating geometric interpretations. In complex analysis, \(e^z\) where \(z = x + iy\) (with \(x\) and \(y\) being real numbers) represents a combination of exponential growth and rotational behavior. In our case, calculating \(e^{1 + i \frac{\text{π}}{2}}\) requires breaking it down using Euler's formula, yielding \(e \times (0 + i) = ie\). This step is vital to simplifying the hyperbolic sine of a complex argument.