### Example: AnglePath with initial position and specified distances/angles

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/AnglePath.md

Illustrates AnglePath with a specified starting point and varying distances and angles for each step.

```mathematica
>> AnglePath({{1, 1}, 90*Degree}, {{1, 90*Degree}, {2, 90*Degree}, {1, 90*Degree}, {2, 90*Degree}})
{{1, 1}, {0, 1}, {0, -1}, {1, -1}, {1, 1}}
```

--------------------------------

### Package Definition Example

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/BeginPackage.md

This example demonstrates how BeginPackage affects the context path. After starting a new package, the context path is updated. EndPackage then reverts this change.

```mathematica
>> BeginPackage("Test`")

>> $ContextPath
{Test`,System`}

>> EndPackage( )

>> $ContextPath
{Test`,System`,Global`}
```

--------------------------------

### SquareFreeQ Examples

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/SquareFreeQ.md

Examples demonstrating the usage of SquareFreeQ for integer inputs.

```mathematica
>> SquareFreeQ(9)
False
```

```mathematica
>> SquareFreeQ(105)
True
```

--------------------------------

### Example Usage

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/Ratios.md

Demonstrates the usage of the Ratios function with an example.

```APIDOC
## Examples

```
>> Ratios({a,b,c,d,e,f,g,h},5)
{(b^5*d^10*f)/(a*c^10*e^5),(c^5*e^10*g)/(b*d^10*f^5),(d^5*f^10*h)/(c*e^10*g^5)}
```
```

--------------------------------

### Example: Array with function, 2D dimension, and origin

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/matheclipse-core/src/main/resources/doc/functions/Array.md

Demonstrates creating a 2x3 matrix with indices starting from (4, 6).

```mathematica
>> Array(f, {2, 3}, {4, 6})
{{f(4,6),f(4,7),f(4,8)},{f(5,6),f(5,7),f(5,8)}}
```

--------------------------------

### Save Example: Saving a list of functions

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/matheclipse-core/src/main/resources/doc/functions/Save.md

Demonstrates how to save a list of function definitions to a temporary file and then retrieve them using Get.

```APIDOC
### Example: Save a definition with dependent symbol definitions into temporary file

```
>> g(x_) := x^3;

>> g(x_,y_) := f(x,y);

>> SetAttributes(f, Listable);

>> f(x_) := g(x^2);

>> temp = FileNameJoin({$TemporaryDirectory, "savedlist.txt"});Print(temp);

>> Save(temp, {f,g})

>> ClearAll(f,g)

>> "Attributes(f)  

>> {f(2),g(7)}

>> Get(temp)
     
>> {f(2),g(7)}
{64,343}

>> Attributes(f)
{Listable}
```
```

--------------------------------

### Start New Context Definition with Begin

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/Begin.md

Use the Begin function to start a new context definition. This function takes a context name as a string argument. The example demonstrates creating a context named 'mytest`', then switching back to the default context.

```matheclipse
Begin("mytest`")
```

```matheclipse
Context()
```

```matheclipse
mytest`
```

```matheclipse
$ContextPath
```

```matheclipse
End()
```

```matheclipse
mytest`
```

```matheclipse
Context()
```

```matheclipse
Global`
```

```matheclipse
$ContextPath
```

--------------------------------

### Split Examples

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/Split.md

Examples demonstrating the usage of the Split function.

```APIDOC
## Examples

### Example 1: Splitting into identical elements
```
>> Split({x, x, x, y, x, y, y, z})
{{x,x,x},{y},{x},{y,y},{z}}
```

### Example 2: Splitting with a test function (equality)
```
>> Split({x, x, x, y, x, y, y, z}, x)
{{x},{x},{x},{y},{x},{y},{y},{z}}
```

### Example 3: Splitting into increasing or decreasing runs
```
>> Split({1, 5, 6, 3, 6, 1, 6, 3, 4, 5, 4}, Less)
{{1,5,6},{3,6},{1,6},{3,4,5},{4}}

>> Split({1, 5, 6, 3, 6, 1, 6, 3, 4, 5, 4}, Greater)
{{1},{5},{6,3},{6,1},{6,3},{4},{5,4}}
```

### Example 4: Splitting based on the first element
```
>> Split({x -> a, x -> y, 2 -> a, z -> c, z -> a}, First(#1) === First(#2) &)
{{x->a,x->y},{2->a},{z->c,z->a}}
```

### Example 5: Splitting an empty list
```
>> Split({})
{}
```
```

--------------------------------

### Start BeakerX Server

Source: https://github.com/axkr/symja_android_library/wiki/BeakerX-usage

Command to start the BeakerX server, typically used before creating a new notebook.

```bash
beakerx
```

--------------------------------

### Initialize SymjaTalk Examples

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/matheclipse-api/src/main/resources/org/matheclipse/api/indexapi.html

Call the setupExamples() function to initialize the interactive examples for SymjaTalk. This function is typically called after the page has loaded to ensure all necessary components are ready.

```javascript
setupExamples();
```

--------------------------------

### Output of the Print Example

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/Print.md

Shows the expected console output when the example code snippet is executed.

```text
1
2
3
4
5
```

--------------------------------

### Start Symja JSON API Server

Source: https://github.com/axkr/symja_android_library/wiki/API

This command starts the Symja JSON API server using a batch script. Ensure Java 11 is installed and JAVA_HOME is set correctly. The classpath includes all libraries in the 'lib' directory.

```bash
"%JAVA_HOME%\bin\java" -Dfile.encoding=UTF-8 -classpath "lib/*" org.matheclipse.api.SymjaServer
```

--------------------------------

### Do Function Examples

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/matheclipse-core/src/main/resources/doc/functions/Do.md

Illustrative examples of how to use the Do function, including basic iteration, nested loops, and control flow with Break and Continue.

```APIDOC
### Examples

```
>> Do(Print(i), {i, 2, 4})
 | 2
 | 3
 | 4

>> Do(Print({i, j}), {i,1,2}, {j,3,5})
 | {1, 3}
 | {1, 4}
 | {1, 5}
 | {2, 3}
 | {2, 4}
 | {2, 5}
```

You can use `Break()` and `Continue()` inside `Do`:

```
>> Do(If(i > 10, Break(), If(Mod(i, 2) == 0, Continue()); Print(i)), {i, 5, 20})
 | 5
 | 7
 | 9

>> Do(Print("hi"),{1+1})
 | hi
 | hi
```

The [A005132 Recaman's sequence](http://oeis.org/A005132) integer sequence

```
>> a = {1}; Do( If( a[ [ -1 ] ] - n > 0 && Position( a, a[ [ -1 ] ] - n ) == {}, a = Append( a, a[ [ -1 ] ] - n ), a = Append( a, a[ [ -1 ] ] + n ) ), {n, 2, 70} ); a
{1,3,6,2,7,13,20,12,21,11,22,10,23,9,24,8,25,43,62,42,63,41,18,42,17,43,16,44,15,45,14,46,79,113,78,114,77,39,78,38,79,37,80,36,81,35,82,34,83,33,84,32,85,31,86,30,87,29,88,28,89,27,90,26,91,157,224,156,225,155}
```
```

--------------------------------

### PDF Examples

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/matheclipse-core/src/main/resources/doc/functions/PDF.md

Illustrative examples demonstrating the usage of the PDF function with different distributions and values.

```APIDOC
### Examples

``` 
>> PDF(NormalDistribution(n, m)) 
1/(Sqrt(2)*E^((-n+#1)^2/(2*m^2))*m*Sqrt(Pi))&
				
>> PDF(GumbelDistribution(n, m),k)
E^(-E^((k-n)/m)+(k-n)/m)/m

>> Table(PDF(NormalDistribution( ), x), {m, {-1, 1, 2}},{x, {-1, 1, 2}})//N  
{{0.24197,0.24197,0.05399},{0.24197,0.24197,0.05399},{0.24197,0.24197,0.05399}}
```
```

--------------------------------

### Graphics3D Example with Multiple Cylinders and Color

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/Cylinder.md

Shows how to create a 3D graphic with multiple cylinders of radius 1, colored yellow, and defined by specific start and end points.

```mathematica
>> Graphics3D({Yellow, Cylinder({{-1, 0, 0}, {1, 0, 0}, {0, 0, Sqrt(3)}, {1, 1, Sqrt(3}}, 1)})
```

--------------------------------

### Example: Array with function, 2D dimension, and single origin value

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/matheclipse-core/src/main/resources/doc/functions/Array.md

Demonstrates creating a 2x3 matrix where the starting index for both dimensions is 3.

```mathematica
>> Array(f, {2, 3}, 3)
{{f(3, 3), f(3, 4), f(3, 5)}, {f(4, 3), f(4, 4), f(4, 5)}}
```

--------------------------------

### FixedPoint Example: Cosine Convergence

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/FixedPoint.md

Demonstrates finding the fixed point of the Cosine function starting from 1.0. The result is the value where Cos(x) = x.

```mathematica
>> FixedPoint(Cos, 1.0)
0.7390851332151607
```

--------------------------------

### SubtractFrom Example Usage

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/SubtractFrom.md

Demonstrates the usage of the SubtractFrom function with a numerical example.

```mathematica
>> a = 10

>> a -= 2
8    
 
>> a
8    
```

--------------------------------

### Load Package using Get()

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/Get.md

Use the Get function to load a package from a specified file path. This function requires a file system to be available.

```mathematica
Get("path-to-package-file-name")
```

--------------------------------

### Append to File Example

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/matheclipse-core/src/main/resources/doc/functions/OpenAppend.md

This example shows how to write initial content to a file using OpenWrite, then append new content using OpenAppend. The file is closed after each operation.

```mathematica
>> f = FileNameJoin({$TemporaryDirectory, "test_open.txt"})
/tmp/test_open.txt
```

```mathematica
>> str = OpenWrite(f);

>> Write(str, x^2 - y^2)

>> Close(str);
```

```mathematica
>> FilePrint(f)
x^2 - y^2
```

```mathematica
>> str = OpenAppend(f);

>> Write(str, x^2 + y^2)

>> Close(str);
```

```mathematica
>> FilePrint(f)
x^2 - y^2
x^2 + y^2
```

--------------------------------

### Start a New Context Definition with Begin

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/matheclipse-core/src/main/resources/doc/functions/Begin.md

Use Begin() to start a new context definition. This function is useful for creating isolated environments for calculations or definitions.

```matheclipse
Begin("<context-name>")
```

--------------------------------

### InputString Example

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/Input.md

Demonstrates the usage of InputString with a specific input. This example shows how to use InputString to process input, in this case, cubing the input "::".

```java
>> InputString("::")^3
```

--------------------------------

### ArrayRules Examples

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/ArrayRules.md

Examples demonstrating the usage of the ArrayRules function.

```APIDOC
### Examples

```
>> a = {{{0,0},{1,1}},{{0,1},{0,1}}}
{{{0,0},{1,1}},{{0,1},{0,1}}}

>> ArrayRules(a)
{{1,2,1}->1,{1,2,2}->1,{2,1,2}->1,{2,2,2}->1,{_,_,_}->0}
```
```

--------------------------------

### Example: FileHash with StringToStream and MD5

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/FileHash.md

Demonstrates computing the MD5 hash of a string converted to a stream. This example shows a practical application of the FileHash function.

```java
>> FileHash(StringToStream("Hello World"),"MD5") 
235328152096874191772633713977838157797
```

--------------------------------

### File Preparation and Initial Write

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/OpenAppend.md

Prepares a file by joining a directory and filename, then opens it for writing and adds initial content. This sets up the file for subsequent appending.

```mathematica
>> f = FileNameJoin({$TemporaryDirectory, "test_open.txt"})
/tmp/test_open.txt

>> str = OpenWrite(f);

>> Write(str, x^2 - y^2)

>> Close(str);
```

--------------------------------

### FixedPoint Example: Iterative Addition

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/FixedPoint.md

Shows an example of using FixedPoint with an anonymous function for iterative addition. It adds 1 to the starting value 1, for 20 iterations.

```mathematica
>> FixedPoint(#+1 &, 1, 20)
21
```

--------------------------------

### StringFreeQ Example 1

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/StringFreeQ.md

Demonstrates basic usage of StringFreeQ with a simple string and pattern.

```mathematica
>> StringFreeQ("symja", "s" ~~__ ~~"a")
False
```

--------------------------------

### Predefined Constant Usage Example

Source: https://github.com/axkr/symja_android_library/wiki/expression-types

Refer to built-in constants. Names start with an uppercase letter.

```plaintext
Degree, E, Pi, False, True, ...
```

--------------------------------

### PolynomialQ Examples

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/PolynomialQ.md

Demonstrates the usage of PolynomialQ with different expressions and variables. These examples show how the function evaluates to true or false based on whether the expression is a polynomial.

```Symja
>> PolynomialQ(x^2 + 7*x + 6)
True
```

```Symja
>> PolynomialQ(Cos(x*y), Cos(x*y))
True
```

```Symja
>> PolynomialQ(2*x^3,x^2)
False
```

--------------------------------

### Example: Default Alphabet Output

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/Alphabet.md

Demonstrates the output when calling Alphabet() to get the English alphabet.

```text
>> Alphabet()
{a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z}
```

--------------------------------

### Context Examples

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/Context.md

Demonstrates the usage of the Context function with different symbols and shows the current context path.

```mathematica
>> $ContextPath
{System`,Global`}

>> Context(a)
Global`

>> Context(Sin)
System`
```

--------------------------------

### Example: SparseArray Initialization from Rules

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/SparseArray.md

Demonstrates the creation of a SparseArray using a rule-based definition. The output shows the number of elements, dimensions, and the default value.

```mathematica
>> SparseArray({{1, 1} -> 1, {2, 2} -> 2, {3, 3} -> 3, {1, 3} -> 4})
SparseArray(Number of elements: 4 Dimensions: {3,3} Default value: 0)
```

--------------------------------

### Get Inverse Function

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/InverseFunction.md

Returns the inverse function for the symbol `head`. For example, the inverse of `Sin` is `ArcSin`.

```mathematica
>> InverseFunction(Sin)
ArcSin
```

--------------------------------

### Increment Example

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/Increment.md

Shows a practical example of using the increment operator. Assigns 2 to 'a', then increments 'a' and shows the returned value (2), followed by the new value of 'a' (3).

```mathematica
>> a = 2;
>> a
2

>> a = 2;
>> a++
2

>> a
3
```

--------------------------------

### Sign of an Interval

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/Interval.md

Determines the sign of a given interval. This example shows how to get the sign for a negative interval.

```java
Sign(Interval({-43, -42}))
```

--------------------------------

### FixedPointList Example: Cosine Convergence

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/FixedPointList.md

Demonstrates the convergence of `FixedPointList` with the `Cos` function starting from 1.0 for 4 iterations.

```mathematica
>> FixedPointList(Cos, 1.0, 4)
{1.0,0.5403023058681398,0.8575532158463934,0.6542897904977791,0.7934803587425656}
```

--------------------------------

### Creating and Showing a JFrame

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/JavaShow.md

This example demonstrates creating a new `javax.swing.JFrame` and then displaying it using the `JavaShow` function. Ensure that Java classes are loaded and accessible before attempting to create and show them.

```mathematica
>> frame= JavaNew["javax.swing.JFrame", "Simple JFrame Demo"];

JavaShow[frame]
```

--------------------------------

### Example: InputForm for Contexts

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/Contexts.md

Demonstrates the InputForm representation of the `Contexts[]` command, showing the expected output format for a list of contexts.

```mathematica
>> Contexts[] // InputForm
{"Global`","Rubi`","System`"}
```

--------------------------------

### Predefined Function Call Example

Source: https://github.com/axkr/symja_android_library/wiki/expression-types

Call built-in functions. Function names start with an uppercase letter, and arguments are in parentheses.

```plaintext
Sin(0), PrimeQ(13)
```

--------------------------------

### Get System Symbols by Pattern

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/Names.md

Use the Names function with a pattern to retrieve all symbols starting with 'System`'. The result can then be filtered further.

```mathematica
>> sysnames = Names("System`*");

>> Select(sysnames, MemberQ(Attributes(#), OneIdentity) &) // InputForm
```

--------------------------------

### Gaussian Integer Primality Examples

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/PrimeQ.md

Illustrates the use of PrimeQ with GaussianIntegers->True to check for primality of Gaussian integers. Shows examples of composite and prime Gaussian integers.

```mathematica
>> PrimeQ(2, GaussianIntegers->True)
False
```

```mathematica
>> PrimeQ(5+2*I, GaussianIntegers->True)
True
```

--------------------------------

### TimesBy Usage Example

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/TimesBy.md

Demonstrates the usage of the TimesBy function with an example. Initialize a variable and then apply TimesBy for multiplication.

```mathematica
>> a = 10

>> a *= 2
20    

>> a
20    
```

--------------------------------

### Example: AssociationThread with Rules

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/matheclipse-core/src/main/resources/doc/functions/AssociationThread.md

Illustrates creating an association using the rule-based syntax. This example shows how to define key-value mappings directly within the function call.

```mathematica
>> AssociationThread({"U","V"} :> {1,2})
<|U:>1,V:>2|>
```

--------------------------------

### Get Cycles of a Permutation

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/Cycles.md

The PermutationCycles function returns the canonical cycle decomposition of a permutation. This example shows how singletons are handled.

```Symja
>> PermutationCycles({4,2,7,6,5,8,1,3})
Cycles({{1,4,6,8,3,7}})
```

--------------------------------

### Example: Principle Components Calculation (Correlation Method)

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/matheclipse-core/src/main/resources/doc/functions/PrincipleComponents.md

Demonstrates the calculation of principle components using the Correlation method.

```java
>> PrincipalComponents({{90.0, 60, 90},{90, 90, 30},{60, 60, 60},{60, 60, 90},{30, 30, 30}}, Method->"Correlation")
{{
 0.911826,-0.942809,0.440421},
 {1.38323,1.41421,-0.0309834},
 {-0.169031,0.0,-0.169031},
 {0.0666714,-0.942809,-0.404733},
 {-2.1927,0.471405,0.164326}}
```

--------------------------------

### UnsameQ Examples

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/matheclipse-core/src/main/resources/doc/functions/UnsameQ.md

Demonstrates the usage of UnsameQ with different inputs. Any object is the same as itself, and numerical equality is checked.

```mathematica
>> a=! =a
False
```

```mathematica
>> 1=! =1.0
True
```

--------------------------------

### Example: FourierDCTMatrix(3, 1)

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/FourierDCTMatrix.md

Demonstrates the creation of a 3x3 DCT matrix using method 1. The output shows the resulting matrix elements.

```mathematica
>> FourierDCTMatrix(3, 1)
{{1/2,1/2,1/2}, 
 {1,0,-1}, 
 {1/2,-1/2,1/2}}
```

--------------------------------

### Find All Occurrences of a Pattern

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/StringPosition.md

Use StringPosition to get a list of all starting and ending positions where a pattern matches within a given string.

```Symja
StringPosition("string", patt)
```

--------------------------------

### Get Fractional Part of a Number

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/FractionalPart.md

Use FractionalPart to extract the decimal portion of a number. For example, FractionalPart(1.5) returns 0.5.

```mathematica
FractionalPart(number)
```

```mathematica
>> FractionalPart(1.5)
0.5
```

--------------------------------

### Example: Import GraphML Graph

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/Import.md

Demonstrates importing a graph in GraphML format from a local file path and the expected output.

```Java
>> Import("c:\\temp\\dotgraph.graphml", "GraphML") 
Graph({1, 2, 3}, {(1,2), (2,3), (3,1)})
```

--------------------------------

### Basic PrimeQ Examples

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/PrimeQ.md

Demonstrates the basic usage of PrimeQ for various integer inputs, including positive, negative, and composite numbers.

```mathematica
>> PrimeQ(2)
True
```

```mathematica
>> PrimeQ(-3)
True
```

```mathematica
>> PrimeQ(137)
True
```

```mathematica
>> PrimeQ(2 ^ 127 - 1)
True
```

```mathematica
>> PrimeQ(1)
False
```

```mathematica
>> PrimeQ(2 ^ 255 - 1)
False
```

--------------------------------

### Distribute Example

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/Distribute.md

Demonstrates how the Distribute function expands a product of sums into a sum of products.

```mathematica
>> Distribute((a+b)*(x+y+z))
```

--------------------------------

### Get Function Usage Documentation

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/beakerx/SymjaGroovy.ipynb

Display the documentation for a given function, including its signature, description, and examples. This is helpful for learning how to use Symja functions.

```python
usage(Im)
```

--------------------------------

### InterpolatingPolynomial Example

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/InterpolatingPolynomial.md

Demonstrates how to get the polynomial representation for a given data-list using the InterpolatingPolynomial function. The input is a list of coordinate pairs and a symbol for the polynomial variable.

```mathematica
>> InterpolatingPolynomial({{1,7},{3,11},{5,27}},x)
(3/2*x-5/2)*(x-1)+7
```

--------------------------------

### Example: Principle Components Calculation (Covariance Method)

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/matheclipse-core/src/main/resources/doc/functions/PrincipleComponents.md

Demonstrates the calculation of principle components using the default Covariance method.

```java
>> PrincipalComponents({{90.0, 60, 90},{90, 90, 30},{60, 60, 60},{60, 60, 90},{30, 30, 30}})
{{
 34.37098,-13.66927,-10.38202},
 {9.98346,47.68821,1.47161},
 {-3.93481,-2.31599,3.89274},
 {14.69692,-25.24923,9.08167},
 {-55.11655,-6.45371,-4.06399}}
```

--------------------------------

### Example: PearsonCorrelationTest returning TestData

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/PearsonCorrelationTest.md

Demonstrates how to use PearsonCorrelationTest to get the test data for two sample vectors. The result is a list containing the correlation coefficient and the p-value.

```mathematica
>> PearsonCorrelationTest({1, 2, 3, 5, 8}, {0.11, 0.12, 0.13, 0.15, 0.18}, "TestData") 
{1.0,0.0}
```

--------------------------------

### Example: AssociationThread with Lists

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/matheclipse-core/src/main/resources/doc/functions/AssociationThread.md

Demonstrates creating an association by providing separate lists for keys and values. The output shows the resulting association with key-value pairs.

```mathematica
>> AssociationThread({"U","V"},{1,2})
<|U->1,V->2|>
```

--------------------------------

### Gather Examples

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/Gather.md

Demonstrates how the Gather function groups elements. The first example shows default grouping, and the second shows grouping with a custom test (implicit equality for fractions).

```Symja
>> Gather({1, 7, 3, 7, 2, 3, 9})
{{1},{7,7},{3,3},{2},{9}}
```

```Symja
>> Gather({1/3, 2/6, 1/9})
{{1/3,1/3},{1/9}}
```

--------------------------------

### DirectedInfinity Examples

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/DirectedInfinity.md

Demonstrates the usage and output of DirectedInfinity with various inputs. Note that division by DirectedInfinity results in 0.

```mathematica
>> DirectedInfinity(1)
Infinity
```

```mathematica
>> DirectedInfinity()
ComplexInfinity
```

```mathematica
>> DirectedInfinity(1 + I)
DirectedInfinity((1+I)/Sqrt(2))
```

```mathematica
>> 1 / DirectedInfinity(1 + I)
0
```

```mathematica
>> DirectedInfinity(1) + DirectedInfinity(-1)
Indeterminate
```

```mathematica
>> DirectedInfinity(1+I)+DirectedInfinity(2+I)
DirectedInfinity((1+I)/Sqrt(2))+DirectedInfinity((2+I)/Sqrt(5))
```

```mathematica
>> DirectedInfinity(Sqrt(3))
Infinity
```

--------------------------------

### Get System Symbols Matching a Pattern

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/matheclipse-core/src/main/resources/doc/functions/Names.md

Use the Names function with a string pattern to retrieve symbols from the 'System' context. The example then filters these symbols to find those with the 'OneIdentity' attribute.

```mma
>> sysnames = Names("System`*"); 
			
>> Select(sysnames, MemberQ(Attributes(#), OneIdentity) &) // InputForm 
{"And","Composition","Dot","GCD","Intersection","Join","Max","Min","Or","Plus","Power","StringExpression","StringJoin","TensorProduct","Times","Union","Xor"}
```

--------------------------------

### Generate Points on Unit Circle

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/matheclipse-core/src/main/resources/doc/functions/CirclePoints.md

Use CirclePoints with a positive integer to get the specified number of points on the unit circle. For example, CirclePoints(4) returns four points.

```mathematica
CirclePoints(i)
```

```mathematica
>> CirclePoints(4)
{{1/Sqrt(2),-1/Sqrt(2)},{1/Sqrt(2),1/Sqrt(2)},{-1/Sqrt(2),1/Sqrt(2)},{-1/Sqrt(2),-1/Sqrt(2)}}
```

--------------------------------

### LengthWhile Usage Example

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/LengthWhile.md

Returns the number of elements at the start of a list for which a given predicate returns True. The predicate is applied using the '# < 10 &' syntax, meaning 'element less than 10'.

```mathematica
>> LengthWhile({1, 2, 3, 10, 5, 8, 42, 11}, # < 10 &)
3
```

--------------------------------

### Example: SparseArray Initialization with Default Value

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/SparseArray.md

Illustrates creating a SparseArray with explicit rules, dimensions, and a default value. The output confirms the number of elements, dimensions, and the specified default value.

```mathematica
>> SparseArray({{1, 1} -> 1, {2, 2} -> 2, {3, 3} -> 3, {1, 3} -> 4}, Automatic, 0)
SparseArray(Number of elements: 4 Dimensions: {3,3} Default value: 0)
```

--------------------------------

### Example: NestList with FromDigits

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/FromDigits.md

Illustrates a sequence generation using NestList, where each element is converted from base 8 to base 9. The output shows the first 50 iterations starting from 8.

```Symja
>> NestList(FromDigits(IntegerDigits(#, 8), 9) &, 8, 50)
{8,9,10,11,12,13,14,15,16,18,20,22,24,27,30,33,37,41,46,51,57,64,81,100,121,145,"181,221,275,345,433,541,761,1036,1471,2014,2787,3927,5533,8537,13555,21441,34102,60891,103386,185033,329032,651411,1286139,2551404,5654254}
```

--------------------------------

### Example: Full Permutation List

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/Ordering.md

Demonstrates calculating the full permutation list for a given set of numbers.

```java
>> Ordering({1,3,4,2,5,9,6})
{1,4,2,3,5,7,6}
```

--------------------------------

### Example: FindMinimum of Sin(x)

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/FindMinimum.md

Demonstrates finding the minimum of the sine function for a single variable 'x' starting at 0.5. The output shows the minimum value and the corresponding variable assignment.

```mathematica
>> FindMinimum(Sin(x), {x, 0.5})
{-1.0,{x->-1.5708}}
```

--------------------------------

### SlotSequence Usage with Plus Function

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/matheclipse-core/src/main/resources/doc/functions/SlotSequence.md

Demonstrates how SlotSequence (##) is used to pass a sequence of arguments to the Plus function. The first example uses all arguments, while the second uses arguments starting from the second one.

```java
>> Plus(##)& [1, 2, 3]
5
```

```java
>> Plus(##2)& [1, 2, 3]
5
```

--------------------------------

### Example: FindMaximum of Sin(x)

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/matheclipse-core/src/main/resources/doc/functions/FindMaximum.md

Demonstrates finding the local maximum of the sine function for a univariate variable `x` starting at `0.5`. The output shows the maximum value and the corresponding variable assignment.

```mathematica
FindMaximum(Sin(x), {x, 0.5})
```

--------------------------------

### Example: FindMaximum of a Multivariate Function with BOBYQA

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/FindMaximum.md

Demonstrates finding the maximum of a complex multivariate function with constraints using the 'BOBYQA' optimization method. The output includes the maximum value and the variable assignments at that point.

```mathematica
>> FindMaximum({(1-x)^2+100*(y-x^2)^2, x >= -2.0 && 2.0 >= x && y >= -0.5 && 1.5 >= y}, {{x, -1.2}, {y,1.0}}, Method->"BOBYQA")
{2034.0,{x->-2.0,y->-0.5}}
```

--------------------------------

### Alternative Method for Getting Coefficients of a Variable Power

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/matheclipse-core/src/main/resources/doc/functions/Coefficient.md

This provides an alternative syntax to the previous example, specifying the variable and its exponent separately. It yields the same result, returning a combined expression for all terms with x^5.

```mathematica
>> Coefficient(poly, x, 5)
84*c^5*y^2-84*c^5*y*z+21*c^5*z^2
```

--------------------------------

### Load VectorAnalysis Package

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/matheclipse-core/src/main/resources/doc/functions/Get.md

Example of loading the VectorAnalysis.m package from the file system.

```mathematica
<<"VectorAnalysis.m"
```

--------------------------------

### Example: Multivariate Optimization with BOBYQA

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/matheclipse-core/src/main/resources/doc/functions/FindMaximum.md

Finds the local maximum of a multivariate function with constraints using the 'BOBYQA' method. The function is `(1-x)^2+100*(y-x^2)^2`, with bounds on `x` and `y`, starting at `x=-1.2` and `y=1.0`.

```mathematica
FindMaximum({(1-x)^2+100*(y-x^2)^2, x >= -2.0 && 2.0 >= x && y >= -0.5 && 1.5 >= y}, {{x, -1.2}, {y,1.0}}, Method->"BOBYQA")
```

--------------------------------

### Load Package using << Syntax

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/Get.md

The << syntax provides an alternative way to load packages from the file system. Ensure a file system is accessible when using this method.

```mathematica
<<"path-to-package-file-name"
```

```mathematica
<<"VectorAnalysis.m"
```

--------------------------------

### Save Dependent Definitions to a Temporary File

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/matheclipse-core/src/main/resources/doc/functions/Save.md

Saves a function definition along with its dependent symbols' definitions to a temporary file. This example demonstrates saving a function `f` which depends on `g`, then clearing them, and finally retrieving them using `Get`.

```mathematica
>> g(x_) := x^3;

>> g(x_,y_) := f(x,y);

>> SetAttributes(f, Listable);

>> f(x_) := g(x^2);

>> temp = FileNameJoin({$TemporaryDirectory, "savedlist.txt"});Print(temp);

>> Save(temp, {f,g})
 
>> ClearAll(f,g)
  
>> "Attributes(f)  

>> {f(2),g(7)}

>> Get(temp)
     
>> {f(2),g(7)}
{64,343}

>> Attributes(f)
{Listable}
```

--------------------------------

### Save Dependent Definitions to Temporary File

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/Save.md

Saves a function definition along with its dependent symbols to a temporary file. The saved definitions can be retrieved later using Get. This example demonstrates saving and then clearing definitions, followed by retrieving them.

```mathematica
>> g(x_) := x^3;

>> g(x_,y_) := f(x,y);

>> SetAttributes(f, Listable);

>> f(x_) := g(x^2);

>> temp = FileNameJoin({$TemporaryDirectory, "savedlist.txt"});Print(temp);

>> Save(temp, {f,g})
 
>> ClearAll(f,g)
  
>> "Attributes(f)  

>> {f(2),g(7)}

>> Get(temp) 
     
>> {f(2),g(7)}
{64,343}

>> Attributes(f)
{Listable}
```

--------------------------------

### Implement DiagonalMatrix Builtin Function in Java

Source: https://github.com/axkr/symja_android_library/wiki/EvalEngine

Example of implementing a Symja builtin function like DiagonalMatrix by extending AbstractFunctionEvaluator. The `evaluate()` method handles the function's logic, `expectedArgSize()` defines argument constraints, and `setUp()` can configure symbol attributes. Return F.NIL to indicate unevaluated results.

```java
private static class DiagonalMatrix extends AbstractFunctionEvaluator {

    @Override
    public IExpr evaluate(final IAST ast, EvalEngine engine) {
        ...

        ...
        // the return value F.NIL has the special meaning "return unevaluated":
        return F.NIL;
    }

    @Override
    public int[] expectedArgSize(IAST ast) {
      // this function allows one or two arguments
      return ARGS_1_2;
    }

    @Override
    public void setUp(final ISymbol newSymbol) {
      // if necessary set attributes here:
      // newSymbol.setAttributes(ISymbol.FLAT | ISymbol.ONEIDENTITY);
    }

  }
```

--------------------------------

### Begin Function

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/Begin.md

Starts a new context definition by providing a context name. This function is useful for managing different scopes of definitions.

```APIDOC
## Begin

```
Begin("<context-name>")
```

### Description
Starts a new context definition.

### Parameters
#### Path Parameters
- **"<context-name>"** (string) - Required - The name of the new context to define.

### Examples
```
>> Begin("mytest`") 

>> Context()
mytest`

>> $ContextPath
{System`,Global`}

>> End()
mytest`

>> Context()
Global`

>> $ContextPath
{System`,Global`}
```
```

--------------------------------

### SlotSequence Starting from Nth Argument

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/SlotSequence.md

Shows how to use SlotSequence to start from the n-th argument. '##2' indicates starting from the second argument.

```java
>> Plus(##2)& [1, 2, 3]
5
```

--------------------------------

### StieltjesGamma Examples

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/matheclipse-core/src/main/resources/doc/functions/StieltjesGamma.md

Examples demonstrating the usage of the StieltjesGamma function.

```mathematica
>> StieltjesGamma(0)
EulerGamma
```

```mathematica
>> StieltjesGamma(0,a)
-PolyGamma(0,a)
```

--------------------------------

### FullSimplify with Options

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/FullSimplify.md

This demonstrates how to use FullSimplify with additional options to control the simplification process. Options like Assumptions and ComplexFunction can be used.

```mathematica
FullSimplify(expr, option1, option2, ...)
```

--------------------------------

### Symbolic Product with Factorial

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/Product.md

Shows how symbolic products can be simplified using factorial notation.

```java
Product(k, {k, 3, n})
```

--------------------------------

### View All Contexts in InputForm

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/matheclipse-core/src/main/resources/doc/functions/Contexts.md

This example demonstrates how to display all available contexts using the `InputForm` representation in Symja. It shows the typical output format for context listing.

```symja
Contexts[] // InputForm
```

--------------------------------

### Additional examples

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/matheclipse-core/src/main/resources/doc/functions/D.md

Various other examples demonstrating the D function's capabilities.

```APIDOC
## Additional examples

### Examples
```
>> D(2/3*Cos(x) - 1/3*x*Cos(x)*Sin(x) ^ 2,x)//Expand    
1/3*x*Sin(x)^3-1/3*Sin(x)^2*Cos(x)-2/3*Sin(x)-2/3*x*Cos(x)^2*Sin(x)
 
>> D(f(#1), {#1,2})    
f''(#1)   
 
>> D((#1&)(t),{t,4})    
0    
 
>> Attributes(f) = {HoldAll}; Apart(f''(x + x))    
f''(2*x)  
 
>> Attributes(f) = {}; Apart(f''(x + x))    
f''(2*x)  
  
>> D({#^2}, #)
{2*#1}
```
```

--------------------------------

### Initial Expression Setup

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/matheclipse-core/src/main/resources/doc/functions/Share.md

Sets up an initial expression 'people' with nested rules, which will be used to demonstrate the Share function.

```mathematica
>> people = <|236234 -> <|"name" -> "bob", "age" -> 18, "sex" -> "M"|>,253456 -> <|"name" -> "sue", "age" -> 25, "sex" -> "F"|>, 323442 -> <|"name" -> "ann", "age" -> 18, "sex" -> "F"|>|>

<|236234-><|name->bob,age->20,sex->M|>,253456-><|name->sue,age->25,sex->F|>,323442-><|name->ann,age->18,sex->F|>|>
```

--------------------------------

### Example: Fitting a Model and Accessing Properties

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/LinearModelFit.md

Demonstrates fitting a linear model with multiple variables and accessing its properties like BestFit, ParameterErrors, and RSquared.

```mathematica
>> nlm=LinearModelFit({{1, 1, 6.5}, {2, 1, 8.4}, {1, 2, 9.6}, {2, 2, 12.1}, {3, 2, 14.0}},{1, a, b, a*b},{a, b})
```

```mathematica
>>nlm // Normal
```

```mathematica
>> nlm("BestFit")
```

```mathematica
>> nlm("BestFitParameters")
```

```mathematica
>> nlm("EstimatedVariance")
```

```mathematica
>> nlm("FitResiduals")
```

```mathematica
>> nlm("ParameterErrors"
```

```mathematica
>> nlm("RSquared")
```

--------------------------------

### Asymptotic Integration of Sine Function

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/AsymptoticIntegrate.md

Demonstrates the asymptotic integration of the sine function around x=0. This example shows how the function approximates the behavior of sin(x) near zero.

```java
>> AsymptoticIntegrate(Sin(x), x, x -> 0)
-1
```

--------------------------------

### DateString() - Example with Timestamp

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/DateString.md

Example of using DateString with a Unix timestamp.

```APIDOC
## DateString(timestamp)

### Description
Converts a Unix timestamp into a formatted date string.

### Method
N/A (Function call)

### Parameters
- **timestamp** (number) - Required - The Unix timestamp.

### Request Example
```
>> DateString(3155673600)
Sat 01 Jan 2000 00:00:00
```

### Response
Returns the date and time corresponding to the provided Unix timestamp, formatted as a string.
```

--------------------------------

### PowerRange Example

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/PowerRange.md

An example demonstrating the usage of the PowerRange function with symbolic expressions.

```APIDOC
### Example
```
>> PowerRange(1, x^10, Sqrt(x))
{1,Sqrt(x),x,x^(3/2),x^2,x^(5/2),x^3,x^(7/2),x^4,x^(9/2),x^5,x^(11/2),x^6,x^(13/2),x^7,x^(15/2),x^8,x^(17/2),x^9,x^(19/2),x^10}
```
```

--------------------------------

### Example: FourierDSTMatrix with DST-1 Method

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/matheclipse-core/src/main/resources/doc/functions/FourierDSTMatrix.md

Demonstrates the usage of FourierDSTMatrix with dimension 3 and method 1 (DST-1), showing the resulting matrix.

```mathematica
>> FourierDSTMatrix(3, 1)
{{1/2,1/Sqrt(2),1/2},
 {1/Sqrt(2),0,-1/Sqrt(2)},
 {1/2,-1/Sqrt(2),1/2}}
```

--------------------------------

### ChineseRemainder Example 3

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/ChineseRemainder.md

Another example with large numbers and negative remainders.

```mathematica
>> ChineseRemainder({-2,-17}, {284407855036305,47})
9669867071234368
```

--------------------------------

### ChineseRemainder Example 2

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/ChineseRemainder.md

An example demonstrating the ChineseRemainder function with large numbers.

```mathematica
>> ChineseRemainder({1,-15}, {284407855036305,47})
8532235651089151
```

--------------------------------

### Calculate BesselK Example

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/BesselK.md

Example of calculating the BesselK function for a specific input.

```mathematica
>> BesselK(1, 3.6)
0.019795
```

--------------------------------

### Array examples

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/Array.md

Demonstrates various ways to use the Array function for creating lists and matrices with different dimensions and origins.

```mathematica
>> Array(f, 4)
{f(1),f(2),f(3),f(4)}
```

```mathematica
>> Array(f, {2, 3})
{{f(1,1),f(1,2),f(1,3)},{f(2,1),f(2,2),f(2,3)}}
```

```mathematica
>> Array(f, {2, 3}, {4, 6})
{{f(4,6),f(4,7),f(4,8)},{f(5,6),f(5,7),f(5,8)}}
```

```mathematica
>> Array(f, 4)
{f(1), f(2), f(3), f(4)}
```

```mathematica
>> Array(f, {2, 3})
{{f(1, 1), f(1, 2), f(1, 3)}, {f(2, 1), f(2, 2), f(2, 3)}}
```

```mathematica
>> Array(f, {2, 3}, 3)
{{f(3, 3), f(3, 4), f(3, 5)}, {f(4, 3), f(4, 4), f(4, 5)}}
```

```mathematica
>> Array(f, {2, 3}, {4, 6})
{{f(4,6),f(4,7),f(4,8)},{f(5,6),f(5,7),f(5,8)}}
```

--------------------------------

### ArcTan Examples

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/matheclipse-core/src/main/resources/doc/functions/ArcTan.md

Examples demonstrating the usage of the ArcTan function with single and double arguments.

```mathematica
>> ArcTan(1)
Pi/4
```

```mathematica
>> ArcTan(1.0)
0.7853981633974483
```

```mathematica
>> ArcTan(-1.0)
-0.7853981633974483
```

```mathematica
>> ArcTan(1, 1)
Pi/4
```

```mathematica
>> ArcTan(-1, 1)
3/4*Pi
```

```mathematica
>> ArcTan(1, -1)
-Pi/4
```

```mathematica
>> ArcTan(-1, -1)
-3/4*Pi
```

```mathematica
>> ArcTan(1, 0)
0
```

```mathematica
>> ArcTan(-1, 0)
Pi
```

```mathematica
>> ArcTan(0, 1)
Pi/2
```

```mathematica
>> ArcTan(0, -1)
-Pi/2
```

--------------------------------

### Example: PolynomialGCD Calculation

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/PolynomialGCD.md

This example demonstrates calculating the GCD of two specific polynomials.

```mathematica
>> PolynomialGCD(x^2 + 7*x + 6, x^2-5*x-6)
1+x
```

--------------------------------

### Example: Subdivide(5)

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/Subdivide.md

Demonstrates subdividing the default range [0, 1] into 5 intervals.

```mathematica
>> Subdivide(5)
{0,1/5,2/5,3/5,4/5,1}
```

--------------------------------

### Erfc Function Examples

Source: https://github.com/axkr/symja_android_library/blob/master/symja_android_library/doc/functions/Erfc.md

Provides examples of calculating the Erfc function for specific values.

```plaintext
>> Erfc(1.0)
0.15729920705028488
```

```plaintext
>> Erfc(0)
1
```