OpenLLM-France/wikimedia huggingface dataset:
# Technical writing/Making a bad requirement good/AdamHanaExercise
## Terms and Definitions

* Billing Session (referred to as a Session for Bill Results, SFBR)
* Hot-bill
* Personal Account
Prerequisits: The system shall allow customers to have several contracts that may contain several Personal Accounts each.

## Printing Invoices

* PI 01: The system shall automatically print invoices at the end of a Billing Session, be it a regular Billing Session or a Hot-bill.
* PI 02: The system shall allow for the following types of invoice printing:
** Regular
** Reprint
** Hot Billing

### Regular Printing

input: SFBR (session for bill results)
* RP 01: The system shall print invoices for the previous month (Billin Session) for all customers at the beginning of the month.
* RP 02: The system shall print regular invoices in a bulk procedure.
* RP 03: The system shall allow the user to define for which customers invoices will be printed according to a list of Personal Accounts.

### Reprint

* Re 01: The system shall allow for the reprinting of all invoices for particular periods of time (for example, previous year, previous month and so forth).

### Hot Billing

* HB 01: The system shall allow for the immediate printing of selected customers' invoices at any point before the end of the reqular Billing Session (Hot-bills).
* HB 02: At the end of each month, the system shall allow the printing all Hot-bills generated in the Billing Session in one-step procedure.
One customer can have several contracts; each contract can have several personal accounts. So to print invoices of one customer it is necessary to use list of personal accounts. If customer closes contract (usually other day then at date of billing session), then a hot-bill will be created and the regular bill created as well.
At the billing session close there runs a check-up for any hot-bills. For the personal account with a hot-bill there will be created a new shorter billing period for the regular bill, excluding the period of the hot-bill.
# Technical writing/Making a bad requirement good/AlistairRomeoExercise

Back to
System requirements
* sys 010 Will print invoices based on previous month activity according to each user account
* sys 020 Must allow bulk printing
* sys 030 Must allow for customer defined printing
* sys 040 Must allow printing of invoice based on date
* sys 050 Must be able to identify which accounts need printing
* sys 060 Must allow up-to-the-moment invoicing
Parameters
* p 010 SFBR
* p 020 personal accounts
* p 030 date range
* p 040 previously printed invoices
* p 050 subscribers
# Technical writing/Making a bad requirement good/BobrobExercise

I010 All clients must print invoices for the previous month.
I020 Invoices must be printed from a list of Personal accounts.
I030 Must have the option to select and print invoices. I040 Must be able to use bulk printing including.
* Hot-billing the bill includes
** date period
** accounts
* Selected billing period
I050 Must be able to reprint all invoices from a specified period.
I060 Must be able to create a hot-bill.
I070 Must be able to vary the billing period.
I080 Must be able to flag hot-bills at any time I090 Must have ability to separate hot billing from regular monthly accounts
At the billing session close there runs a check-up for any hot-bills. For the personal account with a hot-bill there will be created a new shorter billing period for the regular bill, excluding the period of the hot-bill. At the end of each month Client Alpha needs to print all hot-bills in one bulk procedure.
To fully satisfy these needs, it is necessary to create three new methods, which will extend the actual system. These new methods will be called on DataBase level via DataBase link. Because it is impossible to transmit sys_refcursor via DataBase link, “types” should be used.
# Technical writing/Making a bad requirement good/WlodekZdenkaExercise

Invoice requirements
I01. The system must print the invoice on the specified date of the month by default/the date is optional
I02. The system must print the invoice for each customer/ optionally 1 customer, default ALL I03. The system must be able to print the invoice for the particular subscribers/optionally, default ALL I04. The printed period must be changeble/could be set by default
Printing requirements
P01. The invoicing period
P02. Databases of customers,subsribers inc., accounts database P03. The invoice is printed for the first time or reprinted/optionally, default 1.
P04. The session is regular/hot-billing /optionally, default regular
P05. Hot-billing session must be performed after the regular session / flag after the regular session closed
At the beginning of each month all clients (Customer Alpha, Customer Omega) print invoices for previous month. Customer Alpha uses bulk printing. As an input parameter is a SFBR (session for bill result). Sometimes Customer Alpha needs to print invoices only for particular subscribers. Invoices to be printed are chosen by a list of Personal accounts. Sometimes it is needed to reprint all invoices for a particular period of time (i.e. last year etc.). One customer can have several contracts; each contract can have several personal accounts. So to print invoices of one customer it is necessary to use list of personal accounts. If customer closes contract (usually other day then at date of billing session), then a hot-bill will be created and the regular bill created as well. At the billing session close there runs a check-up for any hot-bills. For the personal account with a hot-bill there will be created a new shorter billing period for the regular bill, excluding the period of the hot-bill. At the end of each month Client Alpha needs to print all hot-bills in one bulk procedure.
To fully satisfy these needs, it is necessary to create three new methods, which will extend the actual system. These new methods will be called on DataBase level via DataBase link. Because it is impossible to transmit sys_refcursor via DataBase link, “types” should be used
# Technical writing/Making a bad requirement good/JanaPetrExercise

Invoices:
I-010: Customers must print invoices at the beggining of each month for the previous month
I-020: The system must enable bulk invoice printing for:
* Invoices
* Hot-bills
I-030: The system prints an invoice or a hot-bill for one date/time range
Parameters:
P-040: The system prints invoices according:
* Session for bill result (SFBR)
* Personal accounts
* Period of time
* Contracts:
** All
** Range of contracts:
*** Date/time range
*** Personal accounts
P-050: The system prints hot-bills according:
* Session for bill result (SFBR)
* Personal accounts
* Period of time
* Contracts
At the beginning of each month all clients (Customer Alpha, Customer Omega) print invoices for previous month. Customer Alpha uses bulk printing. As an input parameter is a SFBR (session for bill result). Sometimes Customer Alpha needs to print invoices only for particular subscribers. Invoices to be printed are chosen by a list of Personal accounts. Sometimes it is needed to reprint all invoices for a particular period of time (i.e. last year etc.). One customer can have several contracts; each contract can have several personal accounts. So to print invoices of one customer it is necessary to use list of personal accounts. If customer closes contract (usually other day then at date of billing session), then a hot-bill will be created and the regular bill created as well. At the billing session close there runs a check-up for any hot-bills. For the personal account with a hot-bill there will be created a new shorter billing period for the regular bill, excluding the period of the hot-bill. At the end of each month Client Alpha needs to print all hot-bills in one bulk procedure.
To fully satisfy these needs, it is necessary to create three new methods, which will extend the actual system. These new methods will be called on DataBase level via DataBase link. Because it is impossible to transmit sys_refcursor via DataBase link, “types” should be used.
# Technical writing/Making a bad requirement good/YanaMartinExercise

System performance:
* SP-010: The system must enable bulk printing of invoices for all customers at the beginning of the next month.
* SP-020: The system shall contain a visible list of accounts.
* SP-021: Each contract can have several personal accounts.
* SP-022: One customer can have several contracts.
* SP-030: The system must enable the authorized person to choose accounts for which the invoices will be printed.
* SP-040: The system must enable the authorized person to enter date ranges for which the invoices will be printed.
* SP-050: The system must automacically create a hot-bill and a regular bill when the customer closes the contract.
* SP-051: The system must display and enable to print all hot-bills at the end of each month.
* SP-052: The hot bill report must contain these pieces of information: <To be discussed with Product management and Personal resources management>
System administration:
* SA-010: Only authorized person can maintain the system.
* SA-020: The authorized person must have his/her personal account.
* SA-021: The personal account can be accessed through a nick/name and password.
# RuralWeb/Partners

| Organisation | Status | Country | Remarks |
| Single Spark | potential partner | Netherlands | Single Spark is going to make starter kits for startups in developing countries |
| Soft Productions Africa | potential partner | Ghana | establised part of pilot project, working at ICT-education & making beautiful websites |
| Uppereast ICT Centre | potential partner | Ghana | working at ICT-education and social business |
| Barr ICT Youth Network | potential partner | Uganda | working at ICT-education and social business |
| job-coaching | potential partner | Sint Maarten | ICT-education on job-coaching |
# RuralWeb/Planning

* Need to plan a skype meeting on possible All Voices Count project in cooperation with MDF West Africa
* Who is going to lead in the ICT-working group (online)?
* 14.01.14 Skype meeting ICT department of Groningen University
* 06.01.14 Meeting about Making Beautiful Websites
* December/January: Fundraising for Ruralwebpilots (learning circles ICT-education & entrepreneurship)
* 16.12.13 Inemarie had a meeting with Text to Change
* 16.11.13 Inemarie pitched RuralWeb at Cordaid (http://www.cordaid.org/en/), unfortunately RuralWeb didn't won an award
* 14.11.13 meeting in Nieuwegein, you can also join online

## Goals from 2014 and beyond

Early 2014 (duration 6 months) Pilot phase: implement the 5 small pilot programme plans that are building on the strength of each ICT-partner & will create more content/ e-courses; users; manuals; and a website.
* We need €12.500 to be able to start 5 small pilot projects in 3 countries/with 4 partners.
* RuralWeb will be a project under my own company (www.svikaworks.nl ) & test, adjust, improve RuralWeb (more and improved content, manuals, learning circles, vision, business plan, etc);
* if we can show that RuralWeb’s concept is working & we can show some results, it will be much more easy to ask (cooperate) sponsors or investors for financial support);
From 2015 on: In business phase: start to get financial sustainable, and be able to grow to more ICT centres, ‘rural towns’, rural areas and individual users.
From 2017: financial sustainable / making profit; In 2017 RuralWeb reaches out to at least 126.000 users in developing countries; RuralWeb will be financially independent; and will have African management.

## Goals for 2013

(from which 2 were successful already, and 1 still needs to be obtained):
* Establish partnerships with ICT partners in developing countries to start a pilot with;
* Engage more hands and advisors on board – in the field of ICT, didactics, financials - e.g. in the form of an Advisory Board;
* Find finances for the pilot programmes.

## Already achieved

Since this summer RuralWeb managed to:
* Identify with 4 ICT- partners worldwide (Ghana 2x, Uganda and Curacao/Sint Maarten);
* Have some Dutch volunteers with expertise on ICT; education and/or development cooperation on board;
* Come to pilot project programmes with the ICT-partners;
* Developed a social business plan – which was positively assessed by an expert from Ernst&Young and a ICT-entrepreneur from Kenya – including a market research amongst both suppliers as potential customers in Ghana and Nigeria;
* Test the concept of the business plan at www.facebook.com/RuralWebFAQ (ongoing);
* Got to an online community of currently 150 users and supporters (at Facebook: both RuralWeb and RuralWebFAQ);
* Got publicity at a national Dutch newspaper;
# RuralWeb/KNVB

KNVB is the Dutch Football Association (see KNVB on wikipedia for more info).
In november 2013 we are trying to get in contact with the KNVB. See Dutch wikiversity > RuralWeb > voetbalsite for more info.

## Questions

* What does the KNVB regarding CSR (Corporate Social Responsility)?
** The KNVB is involved in http://www.clublinking.nl/, football players form Holland
** In november 2013 we contacted Bas Stoker, Bas is responsilbe for CSR at the KNVB.
# RuralWeb/Kabissa

On 30.11.13 User:Timboliu sent the following mail to Kabissa: Hello, Erik Hommel told me about Kabissa. I'm an IT consultant working in the Netherlands. Just recently I got in contact with Inemarie Dekker from RuralWeb. We are also stimulating sharing IT knowledge in developing countries. We use Facebook and since recently wikiversity to share knowledge (https://en.wikiversity.org/wiki/RuralWeb). I would love to hear what Kabissa is doing. Regards, Tim Ruijters. Kabissa refers to the website for more info.

## Possible next actions

* Check out the website for more information about current projects.
* Send a mail if Kabissa can help with the pilot project (TD: create page for pilot project)
# Physics equations/14-Heat and Heat Transfer

Q:SpecificHeatEnergyConductivity
# Physics equations/16-Oscillatory Motion and Waves
## Wikiquizzes

Q:amplitudes does not require calculus.
Q:CALCULUS requires calculus and is appropriate only in a calculus-based physics course.
How_things_work_college_course/Waves_(Physics_Classroom)

##### work in progress:

* CALCULUS (advanced material and only in draft form) Physics equations/Oscillations,_waves,_and_interference
# Physics equations/13-Temperature, Kinetic Theory, and Gas Laws/Q:rmsMomentumTransfer/Testbank

Study: http://en.wikiversity.org/wiki/Physics_equations/Sheet/All_chapters
# Physics equations/15-Thermodynamics

Q:heatEngine
# Physics equations/13-Temperature, Kinetic Theory, and Gas Laws/Q:rmsMomentumTransfer

Equations for this quiz Solutions (pdf) Testbank
# Physics equations/14-Heat and Heat Transfer/Q:SpecificHeatEnergyConductivity/Testbank

Study: http://en.wikiversity.org/wiki/Physics_equations/Sheet/All_chapters
# Physics equations/12-Fluid dynamics/Q:pipeDiameterChange/testbank

Study: http://en.wikiversity.org/wiki/Physics_equations/Sheet/All_chapters
# Physics equations/13-Temperature, Kinetic Theory, and Gas Laws

Q:rmsMomentumTransfer
# Physics equations/14-Heat and Heat Transfer/Q:SpecificHeatEnergyConductivity

The accuracy of these answers have not yet been verified.
Solutions (pdf) Equations for this quiz Testbank
# Physics equations/15-Thermodynamics/Q:heatEngine

Testbank Solutions to many wikiquiz questions (pdf) Equations for this quiz
All questions refer to this PV diagram:
# University of Florida/Egm4313/s12.team8.dupre/R2.7

R2.7

## Problem Statement

Develop the Maclaurin series (Taylor Series at t=0) for $\displaystyle e^{t},cos(t),sin(t)$
a) $\displaystyle e^{t}$ (7-1)
b)$\displaystyle cos(t)$ (7-2)
c)$\displaystyle sin(t)$ (7-3)

### Solution

For a Taylor series at any point t, the general form is as follows (to make it more clear, I have changed the usual t in (t-a) to a capital T):
$\sum _{n=0}^{\infty }{\frac {f^{n}(a)}{n!}}(T-a)^{n}=f(a)+{\frac {f'(a)}{1!}}(T-a)+{\frac {f^{2}(a)}{2!}}(T-a)^{2}+{\frac {f^{3}(a)}{3!}}(T-a)^{3}+{\frac {f^{4}(a)}{4!}}(T-a)^{4}...(7-4)$

#### Part a solution

Using (7-4) as applied to (7-1) at time t=0 gives us the Maclaurin series for (7-1):
$\displaystyle e^{t}=\sum _{n=0}^{\infty }{\frac {T^{n}}{n!}}=e^{0}+{\frac {e^{0}}{1!}}(T)+{\frac {e^{0}}{2!}}(T)^{2}+{\frac {e^{0}}{3!}}(T)^{3}+{\frac {e^{0}}{4!}}(T)^{4}...(7-6)$
Simplifying this result, we obtain our final solution:

#### Part b solution

Using (7-4) as applied to (7-2) at time t=0 gives us the Maclaurin series for (7-2):
$\displaystyle cos(t)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}T^{2n}}{(2n)!}}=cos(0)+{\frac {-sin(0)}{1!}}(T)+{\frac {-cos(0)}{2!}}(T)^{2}+{\frac {sin(0)}{3!}}(T)^{3}+{\frac {cos(0)}{4!}}(T)^{4}+...(7-8)$
Simplifying this result, we obtain our final solution:

#### Part c solution

Using (7-5) as applied to (7-3) at time t=0 gives us the Maclaurin series for (7-3):
$\displaystyle sin(t)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}T^{2n+1}}{(2n+1)!}}=sin(0)+{\frac {cos(0)}{1!}}(T)+{\frac {-sin(0)}{2!}}(T)^{2}+{\frac {-cos(0)}{3!}}(T)^{3}+{\frac {sin(0)}{4!}}(T)^{4}+...(7-10)$
Simplifying this result, we obtain our final solution:
# University of Florida/Egm4313/s12.team8.dupre/R3.4

Problem 3.4

## Problem Statement

Use the Basic Rule and the Sum Rule to show that the appropriate particular solution to
$\displaystyle y''-3y'+2y=4x^{2}-6x^{5}$ (4.1)
is of the form
$\displaystyle y_{p}(x)=\sum _{j=0}^{n}c_{j}x^{j}$
with n=5, i.e, $\displaystyle y_{p}(x)=\sum _{j=0}^{5}c_{j}x^{j}$.

### Solution

The Basic rule and Sum rule allow us to choose our particular y forms from table 2.1 on page 82 in the Kreyszig Advanced Engineering Mathematics book. These rules state that, using your given $\displaystyle r(x)$ , you can find what $\displaystyle y_{p}(x)$ you should choose to use.
From (4.1), we know that:
$\displaystyle r(x)=4x^{2}-6x^{5}$ (4.2)
Referring to table 2.1, and knowing that the Basic rule tells us that we can match up the form of our $\displaystyle r(x)$ to one in the table, and use the $\displaystyle y_{p}(x)$ accordingly, we find the following two equations:
The $\displaystyle y_{p}(x)$ for $\displaystyle 4x^{2}$ is:
$\displaystyle k_{2}x^{2}+k_{1}x^{1}+k_{0}$ (4.3)
And the $\displaystyle y_{p}(x)$ for $\displaystyle 6x^{5}$ is:
$\displaystyle K_{5}x^{5}+K_{4}x^{4}+K_{3}x^{3}+K_{2}x^{2}+K_{1}x^{1}+K_{0}$ (4.4)
The sum rule tells us that we are able to add these equations, (4.3) and (4.4), to obtain a final solution for $\displaystyle y_{p}(x)$. The solution for this is as follows:
$\displaystyle k_{2}x^{2}+k_{1}x^{1}+k_{0}+K_{5}x^{5}+K_{4}x^{4}+K_{3}x^{3}+K_{2}x^{2}+K_{1}x^{1}+K_{0}$
Adding similar variables gives us:
$\displaystyle K_{5}x^{5}+K_{4}x^{4}+K_{3}x^{3}+(K_{2}+k_{2})x^{2}+(K_{1}+k_{1})x^{1}+(K_{1}+k_{0})$ (4.5)
Since these k and K values are just constants, we can set:
$\displaystyle K_{5}=c_{5}$
$\displaystyle K_{4}=c_{4}$
$\displaystyle K_{3}=c_{3}$
$\displaystyle (K_{2}+k_{2})=c_{2}$
$\displaystyle (K_{1}+k_{1})=c_{1}$
$\displaystyle (K_{1}+k_{0})=c_{0}$
Our FINAL solution for the $\displaystyle y_{p}(x)$ of (4.1) is:
$\displaystyle y_{p}(x)=c_{5}x^{5}+c_{4}x^{4}+c_{3}x^{3}+c_{2}x^{2}+c_{1}x^{1}+c_{0}$ (4.6)
And, as was the original problem statement, equation (4.6) is of the form:
$\displaystyle y_{p}(x)=\sum _{j=0}^{5}c_{j}x^{j}$
# University of Florida/Egm4313/s12.team8.dupre/R4.4.3

R4.4.3

## Problem Statement

Find $\displaystyle y_{n}(x)$ for n=4,7,11 such that:
$\displaystyle y''_{n}+ay'_{n}+by_{n}=r_{n}(x)$ (1)
for x in [.9,3] with the initial conditions found i.e, $\displaystyle y_{n}(x_{1}),y'_{n}(x_{1}).$
Plot $\displaystyle y_{n}(x)$ for n=4,7,11 for x in [.9,3].

### Solution

Using a modified version of the Matlab code used in 4.4.1, shown here below, we can solve our coefficients for each y equation (at the specified n values):
Our previously solved initial conditions are:
$\displaystyle y_{n}(x_{1})=-22.153,y'_{n}(x_{1})=-55.9732$
Using the above code, we can solve for the coefficients and final y equation to be as shown below:
For n=4:
For n=7:
For n=11:
# University of Florida/Egm4313/s12.teamboss

R2
R3
R4
R5
R6
R7
# University of Florida/Egm4313/s12.team8.dupre/R5.4

Problem 5.4

## Problem Statement

(1) Show that:
$\displaystyle y_{p}(x)=\sum _{i=0}^{n}y_{p,i}(x)$ (5.1)
is indeed the overall particular solution of the L2-ODE-VC:
$\displaystyle y''+p(x)y'+q(x)y=r(x)$ (5.2)
with the excitation:
$\displaystyle r(x)=r_{1}(x)+r_{2}(x)+...+r_{n}(x)=\sum _{i=0}^{n}r_{i}(x)$ (5.3)
(2) Discuss the choice of $\displaystyle y_{p}(x)$ in the above table , e.g., for:
$\displaystyle r(x)=kcos(\omega x)$
Why would you need to have both $\displaystyle cos(\omega x),sin(\omega x)$ in $\displaystyle y_{p}(x)$?

### Solution (1)

Using the following equation:
$\displaystyle r_{i}(x)=y_{p,i}''+p(x)y_{p,i}'+q(x)y_{p,i}$ (5.4)
for different r and y values gives us the following:
$\displaystyle r_{1}(x)=y_{p,1}''+p(x)y_{p,1}'+q(x)y_{p,1}$ (5.5)
$\displaystyle r_{2}(x)=y_{p,2}''+p(x)y_{p,2}'+q(x)y_{p,2}$ (5.6)
$\displaystyle r_{3}(x)=y_{p,3}''+p(x)y_{p,3}'+q(x)y_{p,3}$ (5.7)
Now, adding (5.4),(5.5), and (5.6), gives us:
$\displaystyle r_{1}(x)+r_{2}(x)+r_{3}(x)=(y_{p,1}+y_{p,2}+y_{p,3})''+p(x)(y_{p,1}+y_{p,2}+y_{p,3})'+q(x)(y_{p,1}+y_{p,2}+y_{p,3})$ (5.8)
Equation (5.8) shows us that the overall particular solution of (5.2) with excitation (5.3), is in fact, equation (5.1).

### Solution (2)

We know that the given example for an excitation is the periodic excitation:
$\displaystyle r(x)=kcos(\omega x)$
When we decompose a periodic excitation into a Fourier trigonometric series, we find:
$\displaystyle r(x)=a_{0}+\sum _{n=0}^{\infty }[a_{n}cos(n\omega x)+b_{n}sin(\omega x)]$
Since we know that the particular solution should depend on the excitation, we know that for a periodic excitation $\displaystyle r(x)$, we would need both $\displaystyle cos(\omega x),sin(\omega x)$ in $\displaystyle y_{p}(x)$ to obtain the correct particular solution.
# University of Florida/Egm4313/s12.team8.dupre/R6.1

R6.1

## Problem Statement

Given $\displaystyle f(x)=a_{0}+\sum _{n=1}^{\infty }[a_{n}cos(n\omega x)+b_{n}sin(n\omega x)]$:
(1) Find the (smallest) period of $\displaystyle cos(n\omega x)$ and $\displaystyle sin(n\omega x)$.
(2) Show that these functions also have period $\displaystyle p$.
(3) Show that the constant $\displaystyle a_{0}$ is also a periodic function with period $\displaystyle p$.

### Solution (1)

We know that the period of a normal $\displaystyle sin(x)$ or $\displaystyle cos(x)$ is $\displaystyle 2\pi$. When there are values or variables being multiplied by this $\displaystyle x$ variable, the period becomes $\displaystyle 2\pi$ divided by the values or variables. We know that:
$\displaystyle \omega ={\frac {\pi }{L}}={\frac {2\pi }{p}}$ (6.1.1)
Where $\displaystyle p$ is the period.
Using this relation, along with our variable $\displaystyle n$, we can solve for the period of $\displaystyle cos(n\omega x)$ and $\displaystyle sin(n\omega x)$ as follows:
$\displaystyle p={\frac {2\pi }{\tfrac {n\pi }{L}}}$
$\displaystyle p={\frac {L\pi }{n}}$ (6.1.2)
Since the period will be smallest at $\displaystyle n=1$, plugging into equation (6.1.2) shows that the smallest period of $\displaystyle cos(n\omega x)$ and $\displaystyle sin(n\omega x)$ is:

### Solution (2)

We are also given that:
$\displaystyle \omega ={\frac {2\pi }{p}}$ (6.1.4)
Using this relation, we can solve for the period again as follows:
$\displaystyle p={\frac {2\pi }{\frac {n2\pi }{p}}}$
$\displaystyle p={\frac {p}{n}}$ (6.1.5)
We know that the period is smallest at $\displaystyle n=1$ , and plugging this value into (6.1.5) proves that:

### Solution (3)

We know that, starting at 0:
$\displaystyle {\widetilde {a}}_{0}={\frac {1}{2L}}\int _{0}^{2L}f({\widetilde {x}})d{\widetilde {x}}$ (6.1.7)
Where the period is represented by 0 to 2L. We are also given that:
$\displaystyle {\frac {\pi }{L}}={\frac {2\pi }{p}}$ (6.1.8)
Rearranging (6.1.8) allows us to solve for L:
$\displaystyle L={\frac {p}{2}}$ (6.1.9)
Multiplying (6.1.9) by 2 allows us to find the period of $\displaystyle a_{0}$, as follows:
This shows that $\displaystyle a_{0}$ is indeed a periodic function with a period of $\displaystyle p$. This also shows that at any given $\displaystyle x$ value or period throughout the periodic function, $\displaystyle a_{0}$ holds its constant value.