Question

Identify each of the following integrals or expressions as one of the functions of this chapter and then evaluate it using appropriate formulas and/or tables. $$ \frac{d}{d u}(\operatorname{cn} u) $$

Short answer

\(-\operatorname{sn}u \, \operatorname{dn}u\)

Step-by-step solution

Identify the Function

Recognize that \( \operatorname{cn} \) is the Jacobi elliptic cosine function.

Derivative Formula for Jacobi Elliptic Functions

Recall the derivative rule for the Jacobi elliptic cosine function: \( \frac{d}{du}(\operatorname{cn}u) = -\operatorname{sn}u \, \operatorname{dn}u \).

Apply the Formula

Use the provided rule to find the derivative: \( \frac{d}{du}(\operatorname{cn}u) = -\operatorname{sn}u \, \operatorname{dn}u \).

Key concepts

Jacobi Elliptic Cosine Function

Jacobi elliptic functions are a set of basic functions used in various areas of mathematics, particularly in elliptic integral calculations. One of these functions is the Jacobi elliptic cosine function, denoted as \(\text{cn}(u)\). This function is an analogue to the trigonometric cosine function but tailored for elliptic integrals.

The Jacobi elliptic cosine function \(\text{cn}(u)\) is periodic and oscillates similarly to the cosine function. It plays a crucial role in solving problems that involve the analysis of wave patterns, pendulum motions, and many other physical phenomena where elliptic integrals appear. Understanding its properties and derivatives can help simplify complex mathematical integrals and differential equations.

It's essential to understand that Jacobi elliptic functions also have two other key counterparts: \(\text{sn}(u)\), the Jacobi elliptic sine function, and \(\text{dn}(u)\), the delta amplitude function. These functions are interrelated and are often used together in various mathematical expressions.

Derivative Rules

When dealing with Jacobi elliptic functions, knowing the derivative rules is critical for solving differential equations and evaluating integrals. For the Jacobi elliptic cosine function \(\text{cn}(u)\), there is a specific derivative rule:

\[ \frac{d}{du}(\text{cn}(u)) = -\text{sn}(u) \cdot \text{dn}(u) \]

This formula tells us that the derivative of \(\text{cn}(u)\) with respect to \(u\) involves the product of the negative Jacobi elliptic sine function \(\text{sn}(u)\) and the delta amplitude function \(\text{dn}(u)\).

Here are a few simple steps to understand how to apply this derivative rule:

• Identify that the function in question is \(\text{cn}(u)\).


• Recall the derivative formula for \(\text{cn}(u)\).


• Apply the derivative rule directly to find the expression.

For example, if tasked to find the derivative of \(\frac{d}{du}(\text{cn}(u))\), we use the formula directly:

\[ \frac{d}{du}(\text{cn}(u)) = -\text{sn}(u) \cdot \text{dn}(u) \]

This formula-specific knowledge simplifies otherwise complex differentiation tasks involving Jacobi elliptic functions.

Elliptic Function Identities

Elliptic functions are rich with identities that link them together, making it easier to manipulate and simplify complex expressions. Understanding these identities is vital in advanced mathematical problems.

Some of the most useful identities involving Jacobi elliptic functions are:

• The Pythagorean-like identity:
  \[\text{sn}^2(u) + \text{cn}^2(u) = 1 \]


• The product identity:
  \[\text{dn}^2(u) + k^2 \text{sn}^2(u) = 1 \]
  Where \(k\) is the modulus of the elliptic function.


• The addition formulas:
  \[\text{sn}(u+v) = \frac{\text{sn}(u)\text{cn}(v)\text{dn}(v) + \text{sn}(v)\text{cn}(u)\text{dn}(u)}{1 - k^2 \text{sn}^2(u) \text{sn}^2(v)} \]
  \[\text{cn}(u+v) = \frac{\text{cn}(u)\text{cn}(v) - \text{sn}(u)\text{dn}(u)\text{sn}(v)\text{dn}(v)}{1 - k^2 \text{sn}^2(u) \text{sn}^2(v)} \]

These identities are instrumental in simplifying expressions and solving equations that involve elliptical functions. They serve as the backbone for many proofs and enable mathematicians and physicists to derive new results from existing functions.

Knowing these identities by heart or having a reference could significantly ease the complexity involved in calculations with Jacobi elliptic functions.