# §22.16 Related Functions

## §22.16(i) Jacobi’s Amplitude () Function

### ¶ Definition

22.16.1,

where the inverse sine has its principal value when and is defined by continuity elsewhere. See Figure 22.16.1. is an infinitely differentiable function of .

### ¶ Special Values

For the Gudermannian function see §4.23(viii).

## §22.16(ii) Jacobi’s Epsilon Function

### ¶ Integral Representations

For ,

22.16.14

compare (19.2.5). See Figure 22.16.2.

22.16.31.

## §22.16(iii) Jacobi’s Zeta Function

### ¶ Definition

With and as in §19.2(ii) and ,

See Figure 22.16.3. (Sometimes in the literature is denoted by .)

## §22.16(iv) Graphs

Figure 22.16.1: Jacobi’s amplitude function for and . Values of greater than 1 are illustrated in Figure 22.19.1.
Figure 22.16.2: Jacobi’s epsilon function for and . (These graphs are similar to those in Figure 22.16.1; compare (22.16.3), (22.16.17), and the graphs of in §22.3(i).)
Figure 22.16.3: Jacobi’s zeta function for and .