# User:Egm6321.f09.team7/HW7

## Homework Seven: Problem 1

#### Problem Statement

Pg. 37-1 [[1]]

Show that the following relation is true:

 $x\frac12ln(x\frac{1+x}{1-x})-1 = xtanh^{-1}(x)-1$


#### Problem Solution

We can recall from HW 6 Problem 13 the proof of $\frac12ln(\frac{1+x}{1-x}) = tanh^{-1}(x)$[[2]]

From here a few easy steps give us the proof we are looking for.

First recognize that the relation from the last homework above is part of our original relation if we isolate it we have:

$x(\frac12ln(\frac{1+x}{1-x}))-1 = x(tanh^{-1}(x))-1$

we can cross out the relation from the previous homework and are left with:

$x\cancel{(\frac12ln(\frac{1+x}{1-x}))}-1 = x\cancel{(tanh^{-1}(x))}-1$ = $x(1)-1=x(1)-1$

or

 x-1 = x-1


proving the relation is true

## Homework Seven: Problem 2

#### Problem Statement

Pg. 37-1 [[3]]

From the transparencies equation (2) is defined as:

$Q_n(x) = \underline{P}_n(x)tanh^{-1}(x)-2 \sum\nolimits_{j=1,3,5}^{J} \frac{2n-2j+1}{(2n-j+1)j}\underline{P}_{n-j}(x)$

Using (2)show the relationship of the odd or evenness of Q_n to "n"

#### Problem Solution

First we need to make some definitions within the equation number (2)

The function $tanh^{-1}(x)$ is an odd function both from the definition and also as explained on the transparency

Next the previous homework proved that the polynomial $\underline{P}_n(x)$ is an even function when n is even

It follows that the function $\underline{P}_{n-j}(x)$ is odd for even n since n-j would be odd (j always being odd)

we recognize that the order of the functions is $tanh^{-1}(x)$ multiplied by $\underline{P}_n(x)$ then subtracted by $\underline{P}_{n-j}(x)$ defining an odd function as "O" and an even function as "E" we will have the following two possibilities:

for an even n:

E * O - O leads to O - O which finally gives us O

for odd n:

O * O - E leads to E - E which finally gives us E

(The relationships for odd and even functions were explained in the previous homework if further information is needed)

So the final result for Qn is:

 n=odd Qn=even, n=even Qn=odd


## Homework Seven: Problem 3

#### Problem Statement

Pg. 37-1 [[4]]

Plot the polynmial $/underline{P}_{n}$ for n 0 through 4 and $Q_{n}$ for n 0 through 4

## Homework Seven: Problem 5

#### Problem Statement

See attached file contains the Problem Statement!

#### Problem Solution

See attached file contains the solution!

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400px

 I, the copyright holder of this work, hereby release my work worldwide into the public domain. Where this is not legally possible, I grant anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.

## Homework Seven: Problem 6

#### Problem Statement

See attached file contains the Problem Statement!

#### Problem Solution

See attached file contains the solution!

400px

 I, the copyright holder of this work, hereby release my work worldwide into the public domain. Where this is not legally possible, I grant anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.

400px

 I, the copyright holder of this work, hereby release my work worldwide into the public domain. Where this is not legally possible, I grant anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.

400px

 I, the copyright holder of this work, hereby release my work worldwide into the public domain. Where this is not legally possible, I grant anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.