README.md (5687B)
1 <!-- 2 3 @license Apache-2.0 4 5 Copyright (c) 2018 The Stdlib Authors. 6 7 Licensed under the Apache License, Version 2.0 (the "License"); 8 you may not use this file except in compliance with the License. 9 You may obtain a copy of the License at 10 11 http://www.apache.org/licenses/LICENSE-2.0 12 13 Unless required by applicable law or agreed to in writing, software 14 distributed under the License is distributed on an "AS IS" BASIS, 15 WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. 16 See the License for the specific language governing permissions and 17 limitations under the License. 18 19 --> 20 21 # Heaviside Function 22 23 > Evaluate the [Heaviside function][heaviside-function]. 24 25 <section class="intro"> 26 27 The [Heaviside function][heaviside-function] is defined as 28 29 <!-- <equation class="equation" label="eq:heaviside_function" align="center" raw="H(x) = \begin{cases} 1 & \textrm{if}\ x \gt 0 \\ 0 & \textrm{if}\ x \lt 0\end{cases}" alt="Heaviside function."> --> 30 31 <div class="equation" align="center" data-raw-text="H(x) = \begin{cases} 1 & \textrm{if}\ x \gt 0 \\ 0 & \textrm{if}\ x \lt 0\end{cases}" data-equation="eq:heaviside_function"> 32 <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/heaviside/docs/img/equation_heaviside_function.svg" alt="Heaviside function."> 33 <br> 34 </div> 35 36 <!-- </equation> --> 37 38 and is discontinuous at `0`. Depending on the context, the [Heaviside function][heaviside-function] may be defined as a continuous function. To define the [Heaviside function][heaviside-function] such that the function has rotational symmetry, 39 40 <!-- <equation class="equation" label="eq:heaviside_function_half_maximum" align="center" raw="H(x) = \begin{cases} x & \textrm{if}\ x \gt 0 \\ \frac{1}{2} & \textrm{if}\ x = 0 \\ 0 & \textrm{if}\ x \lt 0\end{cases}" alt="Heaviside function half-maximum."> --> 41 42 <div class="equation" align="center" data-raw-text="H(x) = \begin{cases} x & \textrm{if}\ x \gt 0 \\ \frac{1}{2} & \textrm{if}\ x = 0 \\ 0 & \textrm{if}\ x \lt 0\end{cases}" data-equation="eq:heaviside_function_half_maximum"> 43 <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/heaviside/docs/img/equation_heaviside_function_half_maximum.svg" alt="Heaviside function half-maximum."> 44 <br> 45 </div> 46 47 <!-- </equation> --> 48 49 To define the [Heaviside function][heaviside-function] as a left-continuous function, 50 51 <!-- <equation class="equation" label="eq:heaviside_function_left_continuous" align="center" raw="H(x) = \begin{cases} x & \textrm{if}\ x \gt 0 \\ 0 & \textrm{if}\ x \leq 0\end{cases}" alt="Heaviside function left-continuous."> --> 52 53 <div class="equation" align="center" data-raw-text="H(x) = \begin{cases} x & \textrm{if}\ x \gt 0 \\ 0 & \textrm{if}\ x \leq 0\end{cases}" data-equation="eq:heaviside_function_left_continuous"> 54 <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/heaviside/docs/img/equation_heaviside_function_left_continuous.svg" alt="Heaviside function left-continuous."> 55 <br> 56 </div> 57 58 <!-- </equation> --> 59 60 To define the [Heaviside function][heaviside-function] as a right-continuous function, 61 62 <!-- <equation class="equation" label="eq:heaviside_function_right_continuous" align="center" raw="H(x) = \begin{cases} x & \textrm{if}\ x \geq 0 \\ 0 & \textrm{if}\ x \lt 0\end{cases}" alt="Heaviside function right-continuous."> --> 63 64 <div class="equation" align="center" data-raw-text="H(x) = \begin{cases} x & \textrm{if}\ x \geq 0 \\ 0 & \textrm{if}\ x \lt 0\end{cases}" data-equation="eq:heaviside_function_right_continuous"> 65 <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/heaviside/docs/img/equation_heaviside_function_right_continuous.svg" alt="Heaviside function right-continuous."> 66 <br> 67 </div> 68 69 <!-- </equation> --> 70 71 </section> 72 73 <!-- /.intro --> 74 75 <section class="usage"> 76 77 ## Usage 78 79 ```javascript 80 var heaviside = require( '@stdlib/math/base/special/heaviside' ); 81 ``` 82 83 #### heaviside( x\[, continuity] ) 84 85 Evaluates the [Heaviside function][heaviside-function]. 86 87 ```javascript 88 var v = heaviside( 3.14 ); 89 // returns 1.0 90 91 v = heaviside( -3.14 ); 92 // returns 0.0 93 94 v = heaviside( NaN ); 95 // returns NaN 96 ``` 97 98 The `continuity` parameter may be one of the following values: 99 100 - `'half-maximum'`: if `x == 0`, the function returns `0.5`. 101 - `'left-continuous'`: if `x == 0`, the function returns `0.0`. 102 - `'right-continuous'`: if `x == 0`, the function returns `1.0`. 103 104 By default, the function is discontinuous at `0`. 105 106 ```javascript 107 var v = heaviside( 0.0 ); 108 // returns NaN 109 ``` 110 111 To define the [Heaviside function][heaviside-function] as a continuous function, set the `continuity` parameter. 112 113 ```javascript 114 var v = heaviside( 0.0, 'half-maximum' ); 115 // returns 0.5 116 117 v = heaviside( 0.0, 'left-continuous' ); 118 // returns 0.0 119 120 v = heaviside( 0.0, 'right-continuous' ); 121 // returns 1.0 122 ``` 123 124 </section> 125 126 <!-- /.usage --> 127 128 <section class="examples"> 129 130 ## Examples 131 132 <!-- eslint no-undef: "error" --> 133 134 ```javascript 135 var linspace = require( '@stdlib/array/linspace' ); 136 var heaviside = require( '@stdlib/math/base/special/heaviside' ); 137 138 var x = linspace( -10.0, 10.0, 101 ); 139 var i; 140 141 for ( i = 0; i < x.length; i++ ) { 142 console.log( 'H(%d) = %d', x[ i ], heaviside( x[ i ] ) ); 143 } 144 ``` 145 146 </section> 147 148 <!-- /.examples --> 149 150 <section class="links"> 151 152 [heaviside-function]: https://en.wikipedia.org/wiki/Heaviside_step_function 153 154 </section> 155 156 <!-- /.links -->