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Regularization
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Regularization

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is an option for Sum and Product that specifies what type of regularization to use.

Details

  • Regularization affects only results for divergent sums and products.
  • The following settings can be used to specify regularization procedures for sums of the form :
  • "Abel"
    "Borel"
    "Cesaro"
    "Dirichlet"
  • For alternating sums , the setting "Euler" gives .
  • The following setting can be used to specify a regularization procedure for products :
  • "Dirichlet"
  • Regularization->None specifies that no regularization should be used.
  • For multiple sums and products, the same regularization is by default used for each variable.
  • Regularization->{reg1,reg2,…} specifies regularization regi for the i^(th) variable.

Examples

open allclose all

Basic Examples  (3)Summary of the most common use cases

The following sum does not converge:

In[1]:=1
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https://wolfram.com/xid/0e7preua-fvzgow
Out[1]=1

Using Abel regularization will produce a finite value:

In[2]:=2
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https://wolfram.com/xid/0e7preua-zftb8
Out[2]=2

In this case the Abel-regularized sum does not exist:

In[1]:=1
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https://wolfram.com/xid/0e7preua-fzmo8
Out[1]=1

However, the stronger Borel regularization produces a finite value:

In[2]:=2
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https://wolfram.com/xid/0e7preua-cl0k3n
Out[2]=2

A regularized value of a divergent product:

In[1]:=1
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https://wolfram.com/xid/0e7preua-frnkaa
Out[1]=1

Scope  (5)Survey of the scope of standard use cases

Apply Abel regularization to sum a divergent polynomial-exponential series:

In[1]:=1
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https://wolfram.com/xid/0e7preua-doy2x
Out[1]=1

Use Borel regularization to sum a divergent hypergeometric series:

In[1]:=1
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https://wolfram.com/xid/0e7preua-wu0vj
Out[1]=1

Apply Cesaro regularization to sum a divergent trigonometric series:

In[1]:=1
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https://wolfram.com/xid/0e7preua-csg0gt
Out[1]=1

Sum a divergent logarithmic series using Dirichlet regularization:

In[1]:=1
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https://wolfram.com/xid/0e7preua-naw27j
Out[1]=1

Apply Euler regularization to sum a divergent geometric series:

In[1]:=1
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https://wolfram.com/xid/0e7preua-cpn4vn
Out[1]=1

Applications  (1)Sample problems that can be solved with this function

The regularized sum of all the natural numbers is :

In[1]:=1
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https://wolfram.com/xid/0e7preua-c8zgwu
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0e7preua-i2s4jm
Out[2]=2
Wolfram Research (2008), Regularization, Wolfram Language function, https://reference.wolfram.com/language/ref/Regularization.html.
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Wolfram Research (2008), Regularization, Wolfram Language function, https://reference.wolfram.com/language/ref/Regularization.html.

Text

Wolfram Research (2008), Regularization, Wolfram Language function, https://reference.wolfram.com/language/ref/Regularization.html.

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Wolfram Research (2008), Regularization, Wolfram Language function, https://reference.wolfram.com/language/ref/Regularization.html.

CMS

Wolfram Language. 2008. "Regularization." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Regularization.html.

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Wolfram Language. 2008. "Regularization." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Regularization.html.

APA

Wolfram Language. (2008). Regularization. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Regularization.html

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Wolfram Language. (2008). Regularization. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Regularization.html

BibTeX

@misc{reference.wolfram_2025_regularization, author="Wolfram Research", title="{Regularization}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/Regularization.html}", note=[Accessed: 29-March-2025 ]}

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@misc{reference.wolfram_2025_regularization, author="Wolfram Research", title="{Regularization}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/Regularization.html}", note=[Accessed: 29-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_regularization, organization={Wolfram Research}, title={Regularization}, year={2008}, url={https://reference.wolfram.com/language/ref/Regularization.html}, note=[Accessed: 29-March-2025 ]}

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@online{reference.wolfram_2025_regularization, organization={Wolfram Research}, title={Regularization}, year={2008}, url={https://reference.wolfram.com/language/ref/Regularization.html}, note=[Accessed: 29-March-2025 ]}