<Text-field style="Heading 1" layout="Heading 1">Introduction</Text-field>

<Text-field style="Heading 1" family="Arial" layout="Heading 1">The FourierSeries Package by Amir Khanshan</Text-field>

<Text-field style="Heading 2" layout="Heading 2">Initializations</Text-field>

SShyZXN0YXJ0RyUqcHJvdGVjdGVkRw==

QyQtSSV3aXRoRzYiNiNJJnBsb3RzRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlISIi

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

<Text-field style="Heading 2" layout="Heading 2">Analytic Calculations</Text-field>

Given the function

PkkiZkc2Ii1JKnBpZWNld2lzZUclKnByb3RlY3RlZEc2JjJJInhHRiQiIiFGKzFGK0YqRio=

LUkqcGllY2V3aXNlRyUqcHJvdGVjdGVkRzYmMkkieEc2IiIiIUYpMUYpRidGJw==

defined on 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the trigonometric Fourier series would be generated by the fs command as follows.

LUkjZnNHNiI2JkkiZkdGJC9JInhHRiQ7ISIiIiIiSSV0cmlnR0YkSSJGR0Yk

LCYmSSJhRzYiNiMiIiEjIiIiIiIjLUkkU3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiQsJiomJkkiYkdGJTYjSSJuR0YlRiktSSRzaW5HRi02IyooSSNQaUdGLkYpRjZGKUkieEdGJUYpRilGKSomJkYkRjVGKS1JJGNvc0dGLUY5RilGKS9GNjtGKUkpaW5maW5pdHlHRi5GKQ==

NiQvJkkiYUc2IjYjSSJuR0YmLUkqcGllY2V3aXNlRyUqcHJvdGVjdGVkRzYlL0YoIiIhIyIiIiIiIyooLCYhIiJGMClGNEYoRjBGMEkjUGlHRishIiNGKEY3LyZJImJHRiZGJywkKihGNUYwRjZGNEYoRjRGNA==

Zio2I0kiTkc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCZUIyMiIiIiIiMtSSRTdW1HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2JCwmKiYtSSJiR0YlNiNUJUYsLUkkc2luR0YwNiMsJCoqSSNQaUdGMUYsRjlGLFQnRixUKSEiIkYtRixGLComLUkiYUdGJUY4RiwtSSRjb3NHRjBGPEYsRiwvRjk7Riw5JEYsRiVGJTYqSSNhMEdGJUZMSSJuR0YlRk1JInRHRiVGTkkiVEdGJUZP

The Fourier coefficients LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUYvNiVRIm5GJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUYvNiVRIm5GJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw== are now available as functions of the index LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==, as we see from the experiment

LUkiYUc2IjYjIiIk LUkiYkc2IjYjIiIk

LCQqJEkjUGlHJSpwcm90ZWN0ZWRHISIjI0YmIiIq

LCQqJEkjUGlHJSpwcm90ZWN0ZWRHISIiIyIiIiIiJA==

The decompose command applied to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUYvNiVRIm5GJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw== modifies its representation, taking into account the odd-even cycle produced by 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 in the numerator. Thus, we have

LUkqZGVjb21wb3NlRzYiNiRJImFHRiQiIiM=

LUkqcGllY2V3aXNlRyUqcHJvdGVjdGVkRzYmL0kibkc2IiwkSSJrR0YoIiIjIiIhL0YnLCZGKkYrISIiIiIiLCQqJkkjUGlHRiQhIiNGJ0Y0RjQ=

where, of course, the user has to know the appropriate integer (here 2) to include as a parameter.

The fs command assigns to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== the rule for generating partial sums of the Fourier series. For example, we have

LUkiRkc2IjYjIiIk

LCYjIiIiIiIlRiQtSSRTdW1HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JCwmKiopISIiSSJuR0YrRiRJI1BpR0YpRjBGMUYwLUkkc2luR0YoNiMqKEYyRiRGMUYkSSJ4R0YrRiRGJEYwKiYtSSpwaWVjZXdpc2VHRik2JS9GMSIiISNGJCIiIyooLCZGMEYkRi9GJEYkRjIhIiNGMUZCRiQtSSRjb3NHRihGNUYkRiQvRjE7RiQiIiRGJA==

a display considerably improved with

LUkmdmFsdWVHNiI2Iy1JIkZHRiQ2IyIiJA==

LC4jIiIiIiIlRiQqJkkjUGlHJSpwcm90ZWN0ZWRHISIiLUkkc2luRzYkRihJKF9zeXNsaWJHNiI2IyomRidGJEkieEdGLkYkRiRGJComRichIiMtSSRjb3NHRixGL0YkRjMqJkYnRiktRis2IywkRjAiIiNGJCNGKUY6KiZGJ0YpLUYrNiMsJEYwIiIkRiQjRiRGQComRidGMy1GNUY+RiQjRjMiIio=

Thus, the argument 3 provides the partial sum

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

The exponential form of the Fourier series for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRInhGJ0YvRjIvRjNRJ25vcm1hbEYn is obtained with

LUkjZnNHNiI2JUkiZkdGJC9JInhHRiQ7ISIiIiIiSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ=

LUkkU3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqJiZJImNHRic2I0kibkdGJyIiIi1JJGV4cEdGJDYjKihJI1BpR0YlRi5GLUYuSSJ4R0YnRi5GLi9GLTssJEkpaW5maW5pdHlHRiUhIiJGOA==

LyZJImNHNiI2I0kibkdGJS1JKnBpZWNld2lzZUclKnByb3RlY3RlZEc2JS9GJyIiISMiIiIiIiUsJCooSSNQaUdGKiEiI0YnRjQsKCEiIkYvKUY2RidGLyoqXiNGL0YvRjdGL0YzRi9GJ0YvRi9GLyNGLyIiIw==

where now, the coefficients LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUYvNiVRIm5GJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw== are provided as a function. For example, we have

LUkiY0c2IjYjIiIk

LCQqJkkjUGlHJSpwcm90ZWN0ZWRHISIjLCZGJiIiIiomXiMhIiRGKEYkRihGKEYoI0YoIiM9

and, again, the form of the coefficients can be modified by the decompose command, to

LUkqZGVjb21wb3NlRzYiNiRJImNHRiQiIiM=

LUkqcGllY2V3aXNlRyUqcHJvdGVjdGVkRzYmL0kibkc2IiwkSSJrR0YoIiIjKiheIyMiIiJGK0YvSSNQaUdGJCEiIkYnRjEvRicsJkYqRitGMUYvLCQqKCwoRitGLyooXiNGK0YvRjBGLywmRidGLkYuRi9GL0YvKiZeI0YxRi9GMEYvRi9GL0YwISIjRidGPCNGMUYr

Unfortunately, with the parameter exp, the fs command does not provide a function that generates partial sums. A partial sum of the exponential series has to be constructed from primitives, as per the following.

PkkjRjNHNiItSSRzdW1HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2JComLUkiY0dGJDYjSSJuR0YkIiIiLUkkZXhwR0YnNiMqKl4jRjBGMEYvRjBJI1BpR0YoRjBJInhHRiRGMEYwL0YvOyEiJCIiJA==

LDAjIiIiIiIlRiQqKEkjUGlHJSpwcm90ZWN0ZWRHISIjLCZGKUYkKiZeIyIiJEYkRidGJEYkRiQtSSRleHBHNiRGKEkoX3N5c2xpYkc2IjYjKiheIyEiJEYkRidGJEkieEdGMkYkRiQjRiQiIz0qKF4jIyEiIkYlRiRGJ0Y9LUYvNiMqKF4jRilGJEYnRiRGN0YkRiRGJCooRidGKSwmRilGJComXiNGJEYkRidGJEYkRiQtRi82IyooXiNGPUYkRidGJEY3RiRGJCNGJCIiIyooRidGKSwmRilGJComRklGJEYnRiRGJEYkLUYvNiMqKEZFRiRGJ0YkRjdGJEYkRkoqKF4jRiNGJEYnRj0tRi82IyooXiNGS0YkRidGJEY3RiRGJEYkKihGJ0YpLCZGKUYkKiZGNUYkRidGJEYkRiQtRi82IyooRixGJEYnRiRGN0YkRiRGOA==

Conversion to trigonometric form, and comparison with the corresponding partial sum from the trigonometric series is accomplished with either of the two commands

LUkmZXZhbGNHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kjRjNHRic= LUkmdmFsdWVHSShfc3lzbGliRzYiNiMtSSJGR0YlNiMiIiQ=

The fs command displays the formal Fourier series as part of its output. This can be suppressed by assigning the variable showfs the value false. Thus, the assignment

QyQ+SSdzaG93ZnNHNiJJJmZhbHNlRyUqcHJvdGVjdGVkRyEiIg==

leads to

LUkjZnNHNiI2JUkiZkdGJC9JInhHRiQ7ISIiIiIiSSV0cmlnRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0Yk

NiQvJkkiYUc2IjYjSSJuR0YmLUkqcGllY2V3aXNlRyUqcHJvdGVjdGVkRzYlL0YoIiIhIyIiIiIiIyooLCYhIiJGMClGNEYoRjBGMEkjUGlHRishIiNGKEY3LyZJImJHRiZGJywkKihGNUYwRjZGNEYoRjRGNA==

and

LUkjZnNHNiI2JUkiZkdGJC9JInhHRiQ7ISIiIiIiSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ=

LyZJImNHNiI2I0kibkdGJS1JKnBpZWNld2lzZUclKnByb3RlY3RlZEc2JS9GJyIiISMiIiIiIiUsJCooLCghIiJGLylGNEYnRi8qKl4jRi9GL0Y1Ri9JI1BpR0YqRi9GJ0YvRi9GL0Y4ISIjRidGOSNGLyIiIw==

<Text-field style="Heading 2" layout="Heading 2">Graphical Considerations</Text-field>

Graphing Fourier series created with the fs command relies on Maple's plot command. That is how the graph of the partial sum

PkkjZjNHNiItSSZ2YWx1ZUdJKF9zeXNsaWJHRiQ2Iy1JIkZHRiQ2IyIiJA==

appearing in Figure 1 is obtained.

LUklcGxvdEc2IjYkSSNmM0dGJC9JInhHRiQ7ISIiIiIi

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

Figure 1

Graph of partial sum of Fourier series for 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

The rept command in the package creates a data structure that permits Maple's plot command to draw the periodic extension of a function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRInhGJ0YvRjIvRjNRJ25vcm1hbEYnLUkjbW9HRiQ2MFEiLkYnRj0vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkUvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZXLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYn The periodic extension of

SSJmRzYi

LUkqcGllY2V3aXNlRyUqcHJvdGVjdGVkRzYmMkkieEc2IiIiIUYpMUYpRidGJw==

a function whose domain is 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appears in Figure 2.

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\M0`7e4$!#<$""!""!7$$"3[%3G;%4$y4$!#<$""!""!7$$"33+++++++J!#<$""!""!-I'COLOURG6$%*protectedGI(_syslibG6"6&I$RGBG6$%*protectedGI(_syslibG6"$"*++++"!")$""!""!$""!""!-I*THICKNESSG6$%*protectedGI(_syslibG6"6#""#-I(SCALINGG6"6#I,CONSTRAINEDG6$%*protectedGI(_syslibG6"-I+AXESLABELSG6$%*protectedGI(_syslibG6"6$Q"x6"Q!6"-I%VIEWG6$%*protectedGI(_syslibG6"6$;$!"$""!$"#J!""I(DEFAULTG6$%*protectedGI(_syslibG6"

Figure 2 Periodic extension based on rept command

The graph of a periodic extension and partial sum, seen in Figure 3, is constructed with the Maple display command from the plots package.

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Figure 3 Partial sum and periodic extension superimposed