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JSFHJSFH
<Text-field style="Heading 1" layout="Heading 1">Example 1</Text-field>Let S1 define a simple series expansion:PkkjUzFHNiItSSdzZXJpZXNHJSpwcm90ZWN0ZWRHNiUtSSRleHBHRiQ2I0kieEdGJC9GLCIiISIjNQ==print(); # input placeholderit is required to make:Parse the expression into a table.a list of the series coefficients.a vector of the series coefficients.
<Text-field style="Heading 2" layout="Heading 2">Expression parsed into a table</Text-field>QyctSSl1bmFzc2lnbkc2IjYjLkkjVGJHRiUiIiI/KEkiaUdGJUYpRiktSSVub3BzRyUqcHJvdGVjdGVkRzYjSSNTMUdGJUkldHJ1ZUdGLj4mRig2I0YrLUkjb3BHRi42JEYrRjAhIiItSSZwcmludEdGLkYmprint(); # input placeholderObserving the table it is obvious that the odd elements, i=(2n-1), consist of the series' coefficients, while the even elements are the powers.
<Text-field style="Heading 2" layout="Heading 2">List of coefficients</Text-field>Based on the previous article, all that required are three simple steps: to select only the odd elements of the table, to remove the last element (the residue) from the tableto convert it into a listQy0tSSl1bmFzc2lnbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkLkkjVGJHRiguSSNMc0dGKCIiIj8oSSJpR0YoRi5GLiwkLUklbm9wc0dGJjYjSSNTMUdGKCNGLiIiI0kldHJ1ZUdGJj4mRis2I0YwLUkjb3BHRiY2JCwmRjBGNyEiIkYuRjVGQC1GJDYjLiZGKzYjRjFGQD5GLS1JKGNvbnZlcnRHRiY2JEYrLkklbGlzdEdGJkZALUkmcHJpbnRHRiY2I0YtRi4tSSl3aGF0dHlwZUdGKEZOprint(); # input placeholderObviously the 'dot' operation is not defined for a 'list'LUkwZGVsYXlEb3RQcm9kdWN0RzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHNiRGJUkoX3N5c2xpYkdGKDYkSSNMc0dGKEYtprint(); # input placeholder
<Text-field style="Heading 2" layout="Heading 2">Vector of coefficients</Text-field>Converting a list to a vector is a simple operation in MapleQyktSSl1c2Fzc2lnbkc2IjYjLkkidkdGJSEiIj5GKC1JKGNvbnZlcnRHJSpwcm90ZWN0ZWRHNiRJI0xzR0YlJkknVmVjdG9yRzYkRi1JKF9zeXNsaWJHRiU2I0kkcm93R0YlRiktSSZwcmludEdGLTYjRigiIiItSSV0eXBlR0YtNiRGKC5GMQ==print(); # input placeholderA vector operations are defined, e.g.,LUkwZGVsYXlEb3RQcm9kdWN0RzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHNiRGJUkoX3N5c2xpYkdGKDYkSSJ2R0YoRi0=print(); # input placeholderLUklTm9ybUc2IjYkSSJ2R0YkIiIjprint(); # input placeholderLUklTm9ybUc2IjYkSSJ2R0YkSSlpbmZpbml0eUclKnByb3RlY3RlZEc=print(); # input placeholderJSFH
<Text-field style="Heading 2" layout="Heading 2">The short way</Text-field>Maple has many subtle ways to approach a problem, in this case the "PolynomialTools" package comes very handy...Qyg+SSNTMkc2Ii1JKGNvbnZlcnRHJSpwcm90ZWN0ZWRHNiRJI1MxR0YlLkkocG9seW5vbUdGKCIiIi0mSTBQb2x5bm9taWFsVG9vbHNHNiRGKEkoX3N5c2xpYkdGJTYjSTBDb2VmZmljaWVudExpc3RHRiU2JEYkSSJ4R0YlRi0tJkYwNiNJMkNvZWZmaWNpZW50VmVjdG9yR0YlRjVGLQ==print(); # input placeholderJSFHJSFH%;JSFHJSFHJSFH
<Text-field style="Heading 1" layout="Heading 1">Example 2</Text-field>Let LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjUzNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn be some complex expression:PkkjUzNHNiItSSdleHBhbmRHJSpwcm90ZWN0ZWRHNiMqJiwsIiIiRisqJiZJJ2xhbWJkYUdGJDYjRitGK0kieEdGJEYrRisqJEYwIiIjI0YrRjItSSRzaW5HNiRGJ0koX3N5c2xpYkdGJDYjRjBGKy1JJGV4cEdGNkY4RitGKywoRitGKyomJkYuNiNGMkYrLUkkSW50R0Y2NiQqJi1JJmNvZWZmR0YnNiVJI1MyR0YkRjAiIiRGK0Y0RisvRjA7IiIhSSNQaUdGJ0YrRistSSRjb3NHRjZGOEYrRis=print(); # input placeholderwere the integral is written in an inert form, to shoe the potential substitution later, as it is evaluated separatelyJSFHPkkjSTFHNiItSSRpbnRHRiQ2JC1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjSSJ4R0YkL0YuOyIiIUkjUGlHRis=print(); # input placeholderNow it is easy to place the result into the former expressionPkkjUzRHNiItSShhbGdzdWJzR0YkNiQvLUkkSW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiQtSSRzaW5HRis2I0kieEdGJC9GMjsiIiFJI1BpR0YsSSNJMUdGJEkjUzNHRiQ=print(); # input placeholderJSFHJSFHPkkjUzRHNiItSShjb2xsZWN0RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiRGIzckLUkkY29zR0YnNiNJInhHRiQtSSRleHBHRidGLg==print(); # input placeholderJSFHAssuming that for physical reasons 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, than S4 evaluates into:PkkjUzVHNiItSSVldmFsRyUqcHJvdGVjdGVkRzYkSSNTNEdGJDckLyZJJ2xhbWJkYUdGJDYjIiIiIiIhLyZGLTYjIiIjRi8=print(); # input placeholderThis can easily be ploted over the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJiM4NDc2O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg== domain, as well as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJiM4NDY1O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg== domainLUklcGxvdEc2IjYkSSNTNUdGJC9JInhHRiQ7LCRJI1BpRyUqcHJvdGVjdGVkRyEiIkYr%;JSFHLSZJJnBsb3RzRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJLmNvbXBsZXhwbG90M2RHRig2JkkjUzVHRigvSSJ4R0YoOyomXiQhIiJGMiIiIkkjUGlHRiZGMyomXiRGM0YzRjNGNEYzL0kqbnVtcG9pbnRzR0YoIiUrOy9JJWF4ZXNHRihJJmJveGVkR0Yo%;JSFHJSFHTTdSMApJNlJUQUJMRV9TQVZFLzE1MDY3NjU4MFgqJSlhbnl0aGluZ0c2IjYiW2dsISQlISEhIisiKyIiIkYnI0YnIiIjI0YnIiInI0YnIiNDI0YnIiQ/IiNGJyIkPygjRiciJVNdI0YnIiY/LiUjRiciJyEpR09GJg==TTdSMApJNlJUQUJMRV9TQVZFLzE3MDcwMTQ0NFgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIisiKyIiIkYnI0YnIiIjI0YnIiInI0YnIiNDI0YnIiQ/IiNGJyIkPygjRiciJVNdI0YnIiY/LiUjRiciJyEpR09GJg==