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Chapter 11: Q18P (page 559)

Computer plot graphs of sn u, cn u, and dn u, for several values of k, say, for example, $k = \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 0.9, 0.99 .$ .Also plot 3D graphs of sn, cn, and dn as functions of u and k.

Short Answer

Expert verified

The graphs are shown below.

Step by step solution

01

Given Information

The values, $k = \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 0.9, 0.99 .$

02

Definition of elliptic form.

Thegraph is a pictorial representation of the function.

03

Graph of sn u.

The graph is shown below

04

Graph of cn u.

The graph is shown below.

05

Graph of dn u.

The graph is shown below.

Hence, the solution is mentioned below.

The graphs are shown below.

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Most popular questions from this chapter

Computer plot graphs of K(k) and E(k)in (12.3)for k = 0 - 1. Also plot 3Dgraphs of $F (ϕ, k)$ and $E (ϕ, k)$ in (12.1) for k = 0-1and $ϕ$ from $0 - \frac{π}{2}$ and also from $0 - 2 π$ . Warning: Be sure you understand the notation used by your computer program; see text discussion just after (12.3) and Example 1.

The integral $\int_{0}^{\infty} t^{p - 1} e^{- t} d t = Γ' (p, x)$ is called an incomplete $Γ'$ function. [Note that if x = 0, this integral is $Γ' (p)$ .] By repeated integration by parts, find several terms of the asymptotic series for $Γ' (p, x)$ .

Assuming that x is real, show the following relation between the error function and the Fresnel integrals.

$e r f (\frac{1 - i}{\sqrt{2}} . x) = (1 - i) \sqrt{\frac{2}{π}} \int_{0}^{x} (c o s u^{2} + i s i n u^{2}) d u$

Sketch or computer plot a graph of the function $y = e^{- x^{2}}$ .

Use a graph of $s i n^{2} θ$ and the text discussion just before (12.4)to verify the equations (12.4). Note that the area under the $s i n^{2} θ$ graph from $0 - \frac{π}{2}$ and the area from $\frac{π}{2} - π$ are mirror images of each other, and this will be true also for any function of $\sin^{2} θ$ .

See all solutions

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