OpenLLM-France/wikimedia huggingface dataset:
# Engineering Projects/Power projects/Howard Community College/Fall2012/p3550cyr
### Problem Statement

Develop power supply tester

### Team Members

cyr

### Summary

I got the project started. I began to assemble the circuit. I have researched how the project should be started. The board that will be used to mount the heat sinks and power resistors is set. I have also mounted everything on it. Circuit is startfed. Collected all nessary materials except for a 35 ohm potentiometer.

### Poster

Put a graphic in wikimedia, include the graphic here or link to it here. The graphic should be suitable for creating a traditional project poster.

### Story

Thered were many obstacles. In the beggining I had difficulty getting the wooden board attached to the metal casing. It took longer than it should have to get the power resistors on to the board. The heat sinks were not too difficult. I don't think that hot glue was the best choice to mount them with but future testing would probably be good to test it out. I also had some difficulty finding some parts required for the project. There is still the issue of the 35 ohm potentiometer. I still need to locate one. I spent a lot of time rearranging the circuit. The knobs don't seem to fit the potentiometers. I needed to lengthen the wires a bit to have the pots reach the sides of the metal casing. I did not get far enough to test the power resistor tester.

### Decision List

List all formal decisions made with links to their documentation such as a decision tree or decision matrix.

### Material List

Knobs
Potentiometers
Power resistors
resistors
mosfets
screws
wires
bread boards
transistors
hot glue
power supply casing
wooden board
drill/bits
measuring tape

### Software List

N/A

### Time

48hrs 7min

### Tutorials

All projects create new tutorials of technical details future participants are going to want to know. They are going to be separate pages that are linked to here.

### Next Steps

List specific details, advice, describe the current problems that the next team faces associated with the project here.
# Engineering Projects/Power projects/Howard Community College/Spring2012

       p1550KMN
       p2550TK
       p3550K
# Engineering Projects/Power supply
## Project Power Supply

Many projects require power. PC Tower Power supplies can supply 15 amps at 12 volts. The goal is to re-purpose them for use in electrical projects, test them and begin managing batteries and ultimately solar panels.

## Tutorials

old documentation
# Engineering Projects/Power projects/Howard Community College

    Fall2012
    Spring2012
# Engineering Projects/Power projects/Howard Community College/Fall2012

       p3550cyr
# Boundary Value Problems/Lesson 5.1

Click here to return to BVPs Boundary Value Problems

## Lesson Plan

Requirements of student preparation: The student needs to have worked with vectors. If not the student should obtain suitable instruction in vector calculus.
* Subject Area: A review of vectors, vector operations, the gradient, scalar fields ,vector fields, curl, and divergence.
* Objectives: The learner needs to understand the conceptual and procedural knowledge associated with each of the following
** Vectors,
*** Definition of vectors in ${\mathcal {R}}^{n}$ for $n=1,2,3$
*** Vector Operations
** Scalar and vector fields
** Gradient $\nabla$, divergence $\nabla \circ$, curl $\nabla \times$ and covariant derivatives on fields
** Composite operators such as $\nabla \circ \nabla \mathbf {v}$
* Activities: These structures are to help you understand and aid long-term retention of the material.
** Lesson on Vectors, their associated properties and operations that use vectors.
** Lesson on Scalar and Vector fields
** Lesson on Operations on scalar and vector fields
* Assessment: These items are to determine the effectiveness of the learning activities in achieving the lesson objectives.
** Worksheets
** Quizzes
** Challenging extended problems.
** Student survey/feedback
** Web analytics

## Lesson on Vectors

We will be using only real numbers in this course. The set of all real numbers will be represented by ${\mathcal {R}}$.

###### Definition of a scalar:

A scalar is a single real number, $\displaystyle a\in {\mathcal {R}}$. For example $3$ is a scalar.

###### Definition of a real vector:

A real vector, $\displaystyle v$ is an ordered set of two or more real numbers.
For example: $\displaystyle v=(1,2)$ , $\displaystyle w=(5,0,50,-1.25)$ are both vectors. We will use the notation of $\displaystyle v_{i}$ where the lower index $\displaystyle i=1..n$ represents the individual elements of a vector in the appropropriate order.
Ex: The vector $\displaystyle v=(3,-7)$ has two elements, the first element is designated $\displaystyle v_{1}=3$ and the second is $\displaystyle v_{2}=-7$

###### Dimension of a vector:

The dimension of a vector is the number of elements in the vector.
Ex: Dimension of $\displaystyle v=(-1.25,0,-2,-2)$ is $n=4$

###### Vector Operations:

To refresh your memory, for vectors of the same dimension the following are valid operations:
Let $v=(a,b)$ and $u=(c,d)$ for each of the following statements.
* Addition: $\mathbf {v+w} =(a,b)+(c,d)=(a+c)+(b+d)$
Ex: $v=(2,5)$ and $u=(6,1)$ then $v+w=(8,6)$
* Multiplication by a scalar, $\displaystyle k$: $k(v)=k(a,b)=(ka,kb)$
Ex: $k=2$ and $k(-2,4)=2(-2,4)=(-4,8)$
* Cross Product $\mathbf {u} \times \mathbf {v} =\mathbf {w}$
Let $\mathbf {u=(2,3,4){\mbox{ and }}v=(-1,4,-3)}$ then
$\mathbf {u} \times \mathbf {v} =\left[{\begin{array}{ccc}i&j&k\\2&3&4\\-1&4&-3\end{array}}\right]=\mathbf {i} (3(-3)-4^{2})-\mathbf {j} (2(-3)-4(-1))+\mathbf {k} (2(4)-3(-1)$
$\mathbf {u} \times \mathbf {v} =-25\mathbf {i} +\mathbf {j} +11\mathbf {k}$

## Lesson on Scalar and Vector Fields
### Lesson on Solving Boundary Value Problems with Nonhomogeneous BCs

* Watch .
# Boundary Value Problems/Lesson 5.2

Click here to return to BVPs Boundary Value Problems
* Title: One 1D heat equation with several boundary conditions
* Objectives: Specifically what is to be retained by the learner.
** Setup of heat equation
** Solution of heat equation with homogeneous/nonhomogeneous Dirichlet, Neumann and mixed boudary conditions.
* Activities: Content directed at the learner to achieve learning objectives.
** Solve specific homework prooblems.
* Assessment: Determine lesson effectiveness in achieving the learning Objectives.
** Homework and quizzes

## Background
### The derivation of the Heat Equation

Heat Flow: Fourier's Law
What is heat? Heat is a form of energy that is measured in units of degree Celsius.
For a solid heat is associated with vibrational energy of the crystalline structure.
Heat is measured by us in units of degrees, these are related to calories by the definition, one calorie is the amount of heat required to increase the temperature of one gram of water(at one atmosphere of pressure) by one degree Celsius.
# Boundary Value Problems/Introduction to BVPs/IVP-student-1/Why did I drop the "absolute value" operation

return to IVP-student-1
The answer $y$ will always be either $y<0$ or $y>0$ depending on the initial value $y(0)$. Since $y(0)=3$ and the requirement that $y>0$ we have $|y|=y$ for $-\infty <x<\infty$.
return to IVP-student-1
# Boundary Value Problems/Lesson 1

Return to Boundary Value Problems

## Title: Conservation of mass and flux modeling.
### Objectives

* Develop the student's ability to create qualitative elemental model of mass flow process that will allow extrapolation to heat flow process.

### Methods

* Reading material on development of flux model based on conservatio of mass.
** Consider modeling the movement of a small quantity of material $\delta M$ through a incremental surface,$\delta \Omega$ within a unit of time.
* Have student(s) post elemental models using online simulation package.
* Provide examples of modeling heat flow.

### Assessment

* Provide feedback on a student's mode: critique by "experts", comments by other students, providing student with a document that shows them how to determine the validity of their model, a self grading checklist.
* Multipe choice or short answer Quizzes
Our goal is to understand Partial Derivatives and equations that are composed of terms that include PD's. Such equations are called Partial Differential Equations or PDEs. More specifically, our interest is to solve PDEs , that is find a function $f(x,y)$ such that
$a_{1}(x,y){\frac {\partial ^{2}f}{\partial x^{2}}}+a_{2}(x,y){\frac {\partial ^{2}f}{\partial x\partial y}}+a_{3}(x,y){\frac {\partial ^{2}f}{\partial y^{2}}}+a_{4}(x,y){\frac {\partial f}{\partial x}}+a_{5}(x,y){\frac {\partial f}{\partial y}}+a_{6}(x,y)f+a_{7}(x,y)=0$
# Boundary Value Problems/Lesson 4/Periodic function

A periodic function is a function defined the ways that:
f(x) = f(x+n) (n != 0)
If f has a period k, then:
f(x) = f(x+k) = f(x+kn) (n being an integer)
# Boundary Value Problems/Lesson 2

Boundary Value Problems
--First Order Differential Equation -- If the function $F(t,y)$ is continuous in the variable $t\in \{t_{0}-\delta ,t_{0}+\delta \}$ and uniformly Lipschitz in the variable $y$ ${\frac {dy(t)}{dt}}=F(t,y(t))$
# Boundary Value Problems/Lesson 4.1

Return to previous page Boundary_Value_Problems

#### Sturm Liouville and Orthogonal Functions

The solutions in this BVP course will ALL be expressed as series built on orthogonal functions. Understanding that the simple problem $X''+{\lambda }^{2}X=0$ with the boundary conditions $\alpha _{1}X(a)+\alpha _{2}X'(a)=0$ and $\beta _{1}X(b)+\beta _{2}X'(b)=0$ leads to solutions $X(x)$ that are orthogonal functions is crucial. Once this concept is grasped the majority of the work in this course is repetitive.
In the following notes think of the function $\Phi (x)$ as a substitution for $X(x)$.
TO SEE ALL OF THE PAGES DOUBLE CLICK ON THE FIRST PAGE. THEN YOU WILL BE ABLE TO DOWNLOAD NOTES. THESE WILL BE CONVERTED FOR THE WIKI AALD (at a later date)

#### Fourier Series

From the above work, solving the problem: $X''+{\lambda }^{2}X=0$ with the boundary conditions $X(0)=0$ and $X(L)=0$ leads to an infinite number of solutions $X_{n}(x)=\Phi _{n}(x)=sin\left({\frac {n\pi }{L}}x\right)$ . These are eigenfunctions with eigenvalues $\lambda _{n}={\frac {n\pi }{L}}$

## Homework Assignment from Powell's sixth edition Boundary Value Problems page 71.
### Project 1.2

This is a fourier series application problem. You are given the piecewise defined function $f(t)$ shown in the following graph.
The positive unit pulse is 150 μs in duration and is followed by a 100 μs interval where f(t) =0. Then f(t) is a negative unit pulse for 150 μs once again returning to zero. This pattern is repeated every 2860 μs. We will attempt to represent f(t) as a Fourier series,
* Determine the value of the period: Ans. Period is 2860 μs. The time for a complete repetition of the waveform.
* Find the Fourier Series representation: $f(t)=a_{0}+\sum _{n=1}^{\infty }a_{n}cos(n\pi t/a)+b_{n}sin(n\pi t/a)$ .The video provides an explanation of the determining the coefficients $a_{0},a_{n},b_{n}$
. The results are:$a_{0}=0$ $a_{n}={\frac {\sin \left({\frac {15}{143}}\,n\,\pi \right)+\sin \left({\frac {25}{143}}\,n\,\pi \right)-\sin \left({\frac {40}{143}}\,n\,\pi \right)}{n\pi }}$ $b_{n}={\frac {-\left(-1+\cos \left({\frac {15}{143}}\,n\,\pi \right)+\cos \left({\frac {25}{143}}\,n\,\pi \right)-\cos \left({\frac {40}{143}}\,n\,\pi \right)\right)}{n\pi }}$
* Using 100 terms an approximation is;
* Shift $f(t)$ right or left by an amount $b$ such that the resulting periodic function is an odd function. Here is a plot of shifting it to the left half way between the +1 and -1 pulses. This is a shiift of b= 200 μs. The new funnction is $f(t+200)$. A plot follows: . It could also be shifted to the right by 1230 μs, that is $f(t-1230)$ is the new function.
# Boundary Value Problems/Lesson 10

http://en.wikipedia.org/wiki/Spherical_harmonics
# Boundary Value Problems/Lesson 3

Return to Boundary Value Problems
# Computer Support/Mobile Devices

This section of the computer support course covers mobile devices. Lessons include features, connectivity, security, hardware, and synchronization.

## Preparation

Learners should already be familiar with IC$^3$ - Computer Hardware and IC$^3$ - Living Online.

## Lessons

* Laptop Components
* Laptop Features
* Other Devices
* Ports and Accessories

## See Also

* IC$^3$ - Living Online
# Computer Support/Mobile Devices/Laptop Features

This lesson covers laptop features.

## Objectives and Skills

Objectives and skills for the laptop features portion of A+ certification include:
    Given a scenario, use appropriate laptop features.
   * Special function keys
   ** Dual displays
   ** Wireless (on/off)
   ** Cellular (on/off)
   ** Volume settings
   ** Screen brightness
   ** Bluetooth (on/off)
   ** Keyboard backlight
   ** Touch pad (on/off)
   ** Screen orientation
   ** Media options (fast forward/rewind)
   ** GPS (on/off)
   ** Airplane mode
   * Docking station
   * Physical laptop lock and cable lock
   * Rotating / removable screens

## Readings

* Wikipedia: Laptop

## Multimedia

* YouTube: Laptop Features - CompTIA A+ 220-901 - 3.3
# Computer Support/Networking/Media

This lesson covers networking media.

## Objectives and Skills

Objectives and skills for the networking media portion of A+ certification include:
    Identify the various types of network cables and connectors.
   * Fiber
   ** Connectors: SC, ST and LC
   * Twisted Pair
   ** Connectors: RJ-11, RJ-45
   ** Wiring standards: T568A, T568B
   * Coaxial
   ** Connectors: BNC, F-connector
    Compare and contrast the characteristics of connectors and cabling.
   * Fiber
   ** Types (single-mode vs. multi-mode)
   ** Speed and transmission limitations
   * Twisted pair
   ** Types: STP, UTP, CAT3, CAT5, CAT5e, CAT6, CAT6e, CAT7, plenum, PVC
   ** Speed and transmission limitations
   ** Splitters and effects on signal quality
   * Coaxial
   ** Types: RG-6, RG-59
   ** Speed and transmission limitations
   ** Splitters and effects on signal quality

## Readings

* Wikipedia: Optical fiber
** Read Wikipedia: Optical fiber connector
* Wikipedia: Twisted pair
** Read Wikipedia: Modular connector
* Wikipedia: Coaxial cable
** Read Wikipedia: RF connector

## Multimedia

* YouTube: Network Connectors - CompTIA A+ 220-901 - 2.1
* YouTube: Network Cabling - CompTIA A+ 220-901 - 2.2
* YouTube: Calculating Signal Loss - CompTIA A+ 220-901 - 2.2
* Watch Cisco Learning Network: Ethernet Cables (Types)
* Watch Cisco Learning Network: RJ45 Connectors, RJ45 Wiring, Unshielded Twisted Pair (UTP) and More
# Computer Support/Mobile Devices/Laptop Components

This lesson covers laptop components.

## Objectives and Skills

Objectives and skills for the laptop components portion of A+ certification include:
    Install and configure laptop hardware and components.
   * Expansion options
   ** Express card /34
   ** Express card /54
   ** SODIMM
   ** Flash
   ** Ports/Adapters
   *** Thunderbolt
   *** DisplayPort
   *** USB to RJ-45 dongle
   *** USB to WiFi dongle
   *** USB to Bluetooth
   *** USB Optical Drive
   * Hardware/device replacement
   ** Keyboard
   ** Hard Drive
   *** SSD vs. Hybrid vs. Magnetic disk
   *** 1.8 in vs. 2.5 in
   ** Memory
   ** Smart card reader
   ** Optical drive
   ** Wireless card
   ** Mini-PCIe
   ** Screen
   ** DC jack
   ** Battery
   ** Touchpad
   ** Plastics/frames
   ** Speaker
   ** System board
   ** CPU
    Explain the function of components within the display of a laptop.
   * Types
   ** LCD
   *** TN vs .IPS
   *** Fluorescent vs. LED backlighting
   ** OLED
   * Wi-Fi antenna connector/placement
   * Webcam
   * Microphone
   * Inverter
   * Digitizer

## Readings

* Wikipedia: Laptop
* Wikipedia: ExpressCard
* Wikipedia: SO-DIMM
* Wikipedia: Flash memory
** Wikipedia: Thunderbolt (interface)
** Wikipedia: DisplayPort
** Wikipedia: USB adapter
* Wikipedia: Computer keyboard#Laptop-size
* Wikipedia: Hard disk drive#Form factors
* Wikipedia: PCI Express#PCI Express Mini Card
* Wikipedia: Smart Battery
* Wikipedia: Touchpad
* Wikipedia: Liquid-crystal display
** Wikipedia: Twisted Nematic field effect
** Wikipedia: IPS panel
** Wikipedia: LCD television
** Wikipedia: LED-backlit LCD display
* Wikipedia: OLED
* Wikipedia: Webcam
* Wikipedia: Microphone
* Wikipedia: Power inverter
* Wikipedia: Digitizing

## Multimedia

* YouTube: Laptop Expansion Options - CompTIA A+ 220-901 - 3.1
* YouTube: Replacing a Desktop with a Laptop - CompTIA A+ 220-901 - 3.1
* YouTube: Laptop Displays - CompTIA A+ 220-901 - 3.2
# Computer Support/Mobile Devices/Other Devices

This lesson covers other mobile devices.

## Objectives and Skills

Objectives and skills for the other mobile devices portion of A+ certification include:
    Explain the characteristics of various types of other mobile devices.
   * Tablets
   * Smart phones
   * Wearable technology devices
   ** Smart watches
   ** Fitness monitors
   ** Glasses and headsets
   * Phablets
   * e-Readers
   * Smart camera
   * GPS

## Readings

* Wikipedia: Tablet computer
* Wikipedia: Smartphone
* Wikipedia: Smartwatch
* Wikipedia: Wearable computer
* Wikipedia: Phablet
* Wikipedia: E-reader
* Wikipedia: Smart camera
* Wikipedia: Global Positioning System

## Multimedia

* YouTube: Mobile Devices - CompTIA A+ 220-901 - 3.4
# Computer Support/Networking/Connectivity

This lesson covers networking connectivity.

## Objectives and Skills

Objectives and skills for the networking connectivity portion of A+ certification include:
    Compare and contrast Internet connection types, network types, and their features.
   * Internet Connection Types
   ** Cable
   ** DSL
   ** Dial-up
   ** Fiber
   ** Satellite
   ** ISDN
   ** Cellular
   *** Tethering
   *** Mobilehotspot
   ** Line of sight wireless Internet service
   * Network Types
   ** LAN
   ** WAN
   ** PAN
   ** MAN

## Readings

* Wikipedia: Internet access
* Wikipedia: Cable Internet access
* Wikipedia: Digital Subscriber Line
* Wikipedia: Dial-up Internet access
* Wikipedia: Fiber-optic communication
* Wikipedia: Satellite Internet access
* Wikipedia: Integrated Services Digital Network
* Wikipedia: Cellular network
** Wikipedia: Tethering
* Wikipedia: Computer network
** Wikipedia: Local Area Network
** Wikipedia: Wide Area Network
** Wikipedia: Personal Area Network
** Wikipedia: Metropolitan Area Network

## Multimedia

* YouTube: Internet Connection Types - CompTIA A+ 220-901 - 2.7
* YouTube: Network Types - CompTIA A+ 220-901 - 2.7
# Computer Support/Networking/Devices

This lesson covers networking devices.

## Objectives and Skills

Objectives and skills for the networking devices portion of A+ certification include:
    Compare and contrast network architecture devices, their functions, and features.
   * Hub
   * Switch
   * Router
   * Access point
   * Bridge
   * Modem
   * Firewall
   * Patch panel
   * Repeaters/extenders
   * Ethernet over Power
   * Power over Ethernet injector

## Readings

* Wikipedia: Computer network
* Wikipedia: Ethernet hub
* Wikipedia: Network switch
* Wikipedia: Router (computing)
* Wikipedia: Wireless access point
* Wikipedia: Bridging (networking)
* Wikipedia: Modem
* Wikipedia: Firewall (computing)
* Wikipedia: Patch panel
* Wikipedia: Repeater
* Wikipedia: Power-line communication #Home networking (LAN)
* Wikipedia: Power over Ethernet

## Multimedia

* YouTube: Network Devices - CompTIA A+ 220-901 - 2.8
# Computer Support/Networking

This section of the computer support course covers networking. Lessons include media, addressing, protocols, wireless, routers, Internet connectivity, network types, devices, and tools.

## Preparation

Learners should already be familiar with IC$^3$ - Network Fundamentals.

## Lessons

* Media
* Addressing
* Protocols
* Wireless
* Routers
* Connectivity
* Devices
* Tools

## See Also

* IC$^3$ - Network Fundamentals
# Computer Support/Networking/Addressing

This lesson covers networking addressing.

## Objectives and Skills

Objectives and skills for the networking addressing portion of A+ certification include:
    Explain the properties and characteristics of TCP/IP.
   * IPv4 vs. IPv6
   * Public vs. private vs. APIPA/link local
   * Static vs. dynamic
   * Client-side DNS settings
   * Client-side DHCP
   * Subnet mask vs. CIDR
   * Gateway

## Readings

* Wikipedia: IP address
* Wikipedia: Classful network
* Wikipedia: Subnetworking
* Wikipedia: Link-local address
* Wikipedia: Domain Name System
* Wikipedia: Dynamic Host Configuration Protocol
* Wikipedia: Default gateway

## Multimedia

* YouTube: AAn Overview of IPv4 and IPv6 - CompTIA A+ 220-901 - 2.3
* YouTube: TCP/IP Addressing - CompTIA A+ 220-901 - 2.3

## Activities

* Research IPv4 address classes. Build a table of valid public and private IPv4 address ranges. Then search the Internet for 'verify valid ip address'. Create and test various addresses to see if they are valid or invalid. Does the validator you are using correctly identify public, private, multicast, and experimental address ranges?
* Search the Internet for 'what is my ip'. Identify your public IPv4 address and your public IPv6 address, if you have one. Visit ARIN.net:WhoisRWS or your local regional Internet registry and look up the address registration for your IP addresses. Then search the Internet for 'IPv6 test'. Use several websites to test your IPv6 Internet connection.

## See Also

* Computer Networks - Concepts - Addressing
# Applied linear operators and spectral methods/Differentiating distributions 2
## Differentiation of distributions with severe discontinuities

Now we will consider the case of differentiation of some locally continuous integrable functions whose discontinuities are more severe than simple jumps.

### Example 1

Let us first look at the distribution defined by the locally integrable function
    $f(x)=H(x)~\log x$
where $H(x)$ is the Heaviside function.
Then
    $\left\langle f,\phi \right\rangle =\int _{0}^{\infty }\phi (x)~\log(x)~{\text{d}}x$
By definition
    $\left\langle f',\phi \right\rangle =-\left\langle f,\phi '\right\rangle =-\int _{0}^{\infty }\phi '(x)~\log(x)~{\text{d}}x$
Since the right hand side is a convergent integral, we can write
    $\left\langle f',\phi \right\rangle =-\lim _{\epsilon \rightarrow 0}\int _{\epsilon }^{\infty }\phi '(x)~\log x~{\text{d}}x$
Integrating by parts,
    $\left\langle f',\phi \right\rangle =-\lim _{\epsilon \rightarrow 0}\left[\int _{\epsilon }^{\infty }{\cfrac {\phi (x)}{x}}~{\text{d}}x+\phi (\epsilon )~\log(\epsilon )\right]$
Now, as $\epsilon \rightarrow 0$ we have $\phi (\epsilon )\approx \phi (0)$ and therefore
    ${\text{(1)}}\qquad \left\langle f',\phi \right\rangle \approx -\lim _{\epsilon \rightarrow 0}\left[\int _{\epsilon }^{\infty }{\cfrac {\phi (x)}{x}}~{\text{d}}x+\phi (0)~\log(\epsilon )\right]$
The right hand side of (1) gives a meaning to (i.e., regularizes) the divergent integral
    $\int _{0}^{\infty }{\cfrac {\phi (x)}{x}}~{\text{d}}x~.$
We write
    ${\left\langle f',\phi \right\rangle =\left\langle {\text{pf}}\left[{\cfrac {H(x)}{x}}\right],\phi \right\rangle }$
where ${\text{pf}}[\bullet ]$ is the pseudofunction which is defined by the right hand side of (1).
In this sense, if
    $f(x)=H(x)~\log x$
then
    ${f'(x)={\text{pf}}\left[{\cfrac {H(x)}{x}}\right]~.}$

### Example 2

Next let the function to be differentiated be
    $f(x)=\log |x|$
We can write this function as
    $f(x)=H(x)~\log x+H(-x)~\log(-x)~.$
Then
    ${f'(x)={\text{pf}}\left[{\cfrac {H(x)}{x}}\right]+{\text{pf}}\left[{\cfrac {H(-x)}{x}}\right]={\text{pf}}\left[{\cfrac {1}{x}}\right]}$
where the pseudofunction $1/x$ is defined as the distribution
    ${\left\langle {\text{pf}}\left[{\cfrac {1}{x}}\right],\phi \right\rangle =\lim _{\epsilon \rightarrow 0}\left[\int _{\epsilon }^{\infty }{\cfrac {\phi (x)}{x}}~{\text{d}}x+\int _{-\infty }^{\epsilon }{\cfrac {\phi (x)}{x}}~{\text{d}}x\right]}$
The individual terms diverge at $\epsilon \rightarrow 0$ but the sum does not.
In this way we have assigned a value to the usually divergent integral
    $\int _{\infty }^{\infty }{\cfrac {\phi (x)}{x}}~dx~.$
This value is more commonly known as the Cauchy Principal Value.
Template:Lectures