Question

Find all the zeros of \(\sinh z\) and \(\cosh z\).

Short answer

The zeros of \(\sinh z\) are \(z = n\pi i\), for \(n\) an integer. The zeros of \(\cosh z\) are \(z = (2n + 1)\pi i / 2\), for \(n\) an integer.

Step-by-step solution

Define the Hyperbolic Functions

Let's begin by defining the hyperbolic sine and hyperbolic cosine functions. The hyperbolic sine, \(\sinh z\), is defined as \(\sinh z = (e^{z} - e^{-z})/2\) and the hyperbolic cosine, \(\cosh z\), is defined as \(\cosh z = (e^{z} + e^{-z})/2\).

Find Zeroes of \(\sinh z\)

Setting \(\sinh z = 0\) gives the equation \((e^{z} - e^{-z})/2 = 0\). This equation can be rewritten as \(e^{z} = e^{-z}\), or equivalently, \(e^{2z} = 1\). This equation is true for \(z = n\pi i\), where \(n\) is an integer.

Find Zeroes of \(\cosh z\)

Setting \(\cosh z = 0\) gives the equation \((e^{z} + e^{-z})/2 = 0\). This equation simplifies to \(e^{z} = -e^{-z}\), or \(e^{2z} = -1\). This equation is only true when the exponent of \(e\) is an odd integer multiple of \(\pi i\), so the solutions are \(z = (2n + 1)\pi i / 2\), where \(n\) is an integer.

Key concepts

Hyperbolic Sine

The hyperbolic sine function, denoted as \(\sinh z\), is a transcendental function similar in form to the familiar trigonometric sine function, but it pertains to hyperbolic geometry. As defined in complex analysis, \(\sinh z = \frac{e^z - e^{-z}}{2}\), where \(e\) is the base of the natural logarithm and \(z\) is a complex number. This function can be seen as describing the shape of a hanging cable or chain, known as a catenary. Unlike its trigonometric counterpart, \(\sinh z\) is not periodic, meaning it does not repeat values at regular intervals.

Finding zeros of \(\sinh z\) involves solving the equation \(\sinh z = 0\). To understand this better, imagine squaring both sides of the equation and using the identity \(\sinh^2 z + 1 = \cosh^2 z\). Hence, solving \(\sinh z = 0\) is equivalent to finding points where the hyperbolic sine squared plus one equals the hyperbolic cosine squared. The zeros of \(\sinh z\) are pure imaginary numbers and occur at integral multiples of \(\pi i\), forming a sequence of points along the imaginary axis in the complex plane.

Hyperbolic Cosine

Complementing the hyperbolic sine, the hyperbolic cosine function, \(\cosh z\), is defined as \(\cosh z = \frac{e^z + e^{-z}}{2}\). It represents the shape of a flexible string or arch. \(\cosh z\) shares similarities with the cos function in trigonometry but, like \(\sinh z\), it is not periodic and grows exponentially. One of the notable qualities of \(\cosh z\) is that it is an even function, which means that \(\cosh(z) = \cosh(-z)\).To find the zeroes of \(\cosh z\), we observe where the function equals zero. Given \(\cosh z = 0\), we derive the necessary conditions for the exponential terms to cancel each other out. The zeros of \(\cosh z\) arise at the half-odd integral multiples of \(\pi i\), thus occurring exclusively on the imaginary axis but staggered between the zeros of \(\sinh z\). These solutions are critical for understanding the symmetries and properties of hyperbolic functions in complex analysis.

Complex Analysis

Complex analysis is a branch of mathematics that studies functions that operate within the realm of complex numbers. A complex number is of the form \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit, satisfying \(i^2 = -1\).

In complex analysis, concepts like continuity, differentiability, integration, and representation of functions take on fascinating properties that differ from their real number counterparts. Certain functions, known as holomorphic functions, are especially interesting because they are complex-differentiable — an attribute that requires a function to satisfy the Cauchy-Riemann equations and exhibit smooth behavior.

Hyperbolic functions are a fundamental example of functions studied within complex analysis. Their zeros, growth rates, and behaviors offer insight into structural properties of complex functions. For instance, the zero points of \(\sinh z\) and \(\cosh z\) highlight the periodic nature of exponential functions when extended into the complex plane.

Zeroes of Transcendental Functions

Transcendental functions extend beyond polynomial equations and often involve exponential, logarithmic, trigonometric, and hyperbolic functions. Zeroes of transcendental functions are the values at which the function's output equals zero. In the complex plane, these zeroes can be complex numbers and are often fundamental to solving equations in higher mathematics.

For example, hyperbolic functions are transcendental, and their zeroes have intriguing implications in various areas of mathematics and physics. The zeroes of \(\sinh z\) and \(\cosh z\) provide us with concrete solutions to equations that cannot be solved using algebraic methods alone. These zeroes often correspond to critical points of physical systems or conditions for the existence of certain mathematical phenomena. A deep understanding of where these functions become zero aids in the analysis of wave behavior, quantum mechanics, signal processing, and many other fields.