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In `F33f`_`[mathematics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Mathematics]`_`f, `!hypercomplex analysis`! is the extension of `F33f`_`[complex analysis`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Complex_analysis]`_`f to the `F33f`_`[hypercomplex numbers`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Hypercomplex_number]`_`f. The first instance is functions of a `F33f`_`[quaternion variable`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Quaternion_variable]`_`f, where the argument is a `F33f`_`[quaternion`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Quaternion]`_`f (in this case, the sub-field of hypercomplex analysis is called `F33f`_`[quaternionic analysis`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Quaternionic_analysis]`_`f). A second instance involves functions of a `F33f`_`[motor variable`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Motor_variable]`_`f where arguments are `F33f`_`[split-complex numbers`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Split-complex_number]`_`f.

In `F33f`_`[mathematical physics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Mathematical_physics]`_`f, there are hypercomplex systems called `F33f`_`[Clifford algebras`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Clifford_algebra]`_`f. The study of functions with arguments from a Clifford algebra is called `F33f`_`[Clifford analysis`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Clifford_analysis]`_`f.

A `F33f`_`[matrix`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Matrix_(mathematics)]`_`f may be considered a hypercomplex number. For example, the study of functions of 2 × 2 `F33f`_`[real`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Real_number]`_`f matrices shows that the `F33f`_`[topology`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Topological_space]`_`f of the `F33f`_`[space`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Space_(mathematics)]`_`f of hypercomplex numbers determines the function theory. Functions such as `F33f`_`[square root of a matrix`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Square_root_of_a_matrix]`_`f, `F33f`_`[matrix exponential`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Matrix_exponential]`_`f, and `F33f`_`[logarithm of a matrix`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Logarithm_of_a_matrix]`_`f are basic examples of hypercomplex analysis.`:cite-ref-1[`F5bf`_`[1`#cite-note-1]`_`f] The function theory of `F33f`_`[diagonalizable matrices`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Diagonalizable_matrices]`_`f is particularly transparent since they have `F33f`_`[eigendecompositions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Eigendecomposition]`_`f.`:cite-ref-2[`F5bf`_`[2`#cite-note-2]`_`f] Suppose T = ∑ ∑ i = 1 N λ λ i E i {\\displaystyle \\textstyle T=\\sum _{i=1}^{N}\\lambda _{i}E_{i}} where the `*E`*`*i`* are `F33f`_`[projections`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Projection_(linear_algebra)]`_`f. Then for any `F33f`_`[polynomial`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Polynomial]`_`f f {\\displaystyle f} , f ( T ) = ∑ ∑ i = 1 N f ( λ λ i ) E i . {\\displaystyle f(T)=\\sum _{i=1}^{N}f(\\lambda _{i})E_{i}.}

The modern terminology for a "system of hypercomplex numbers" is an `*`F33f`_`[algebra`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Algebra_over_a_field]`_`f over the real numbers`*, and the algebras used in applications are often `F33f`_`[Banach algebras`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Banach_algebra]`_`f since `F33f`_`[Cauchy sequences`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Cauchy_sequence]`_`f can be taken to be `F33f`_`[convergent`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Convergent_sequence]`_`f. Then the function theory is enriched by `F33f`_`[sequences`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Sequence]`_`f and `F33f`_`[series`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Series_(mathematics)]`_`f. In this context the extension of `F33f`_`[holomorphic functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Holomorphic_function]`_`f of a `F33f`_`[complex`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Complex_number]`_`f variable is developed as the `F33f`_`[holomorphic functional calculus`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Holomorphic_functional_calculus]`_`f. Hypercomplex analysis on Banach algebras is called `F33f`_`[functional analysis`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Functional_analysis]`_`f.

>>Contents

• `F0af`_`[See also`#see-also]`_`f

• `F0af`_`[References`#references]`_`f

• `F0af`_`[Sources`#sources]`_`f

-─

>>See also

• `F33f`_`[Giovanni Battista Rizza`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Giovanni_Battista_Rizza]`_`f

• `F33f`_`[Biquaternion functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Biquaternion_functions]`_`f

>>References

`:cite-note-1`!1.`! `F0af`_`[↑`#cite-ref-1]`_`f `F33f`_`[Felix Gantmacher`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Felix_Gantmacher]`_`f (1959) `*The Theory of Matrices`*, two volumes, translator: `F33f`_`[Kurt Hirsch`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Kurt_Hirsch]`_`f, `F33f`_`[Chelsea Publishing`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Chelsea_Publishing]`_`f, chapter 5: functions of matrices, chapter 8: roots and logarithms of matrices

`:cite-note-2`!2.`! `F0af`_`[↑`#cite-ref-2]`_`f Shaw, Ronald (1982) `*Linear Algebra and Group Representations`*, v. 1, § 2.3, Diagonalizable linear operators, pages 78–81, `F33f`_`[Academic Press`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Academic_Press]`_`f `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 0-12-639201-3.

>>>Sources

• Daniel Alpay (ed.) (2006) `*Wavelets, Multiscale systems and Hypercomplex Analysis`*, Springer, `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 9783764375881 .

• Enrique Ramirez de Arellanon (1998) `*Operator theory for complex and hypercomplex analysis`*, `F33f`_`[American Mathematical Society`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=American_Mathematical_Society]`_`f (Conference proceedings from a meeting in Mexico City in December 1994).

• J. A. Emanuello (2015) Analysis of functions of split-complex, multi-complex, and split-quaternionic variables and their associated conformal geometries, Ph.D. Thesis, `F33f`_`[Florida State University`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Florida_State_University]`_`f

• Sorin D. Gal (2004) `*Introduction to the Geometric Function theory of Hypercomplex variables`*, Nova Science Publishers, `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 1-59033-398-5.

• `:citerefl-vi-kao-farrellshort2007`aLávička, Roman; `F33f`_`[O'Farrell, Anthony G.`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=A.G._O’Farrell]`_`f; Short, Ian (2007). "Reversible maps in the group of quaternionic Möbius transformations" (PDF). `*`F33f`_`[Mathematical Proceedings of the Cambridge Philosophical Society`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Mathematical_Proceedings_of_the_Cambridge_Philosophical_Society]`_`f`*. `!143`! (1): 57–69. `F33f`_`[Bibcode`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Bibcode_(identifier)]`_`f:2007MPCPS.143...57L. `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1017/S030500410700028X.

• `F33f`_`[Irene Sabadini`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Irene_Sabadini]`_`f and Franciscus Sommen (eds.) (2011) `*Hypercomplex Analysis and Applications`*, Birkhauser Mathematics.

• Irene Sabadini & Michael V. Shapiro & F. Sommen (editors) (2009) `*Hypercomplex Analysis`*, Birkhauser `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-3-7643-9892-7.

• Sabadini, Sommen, Struppa (eds.) (2012) `*Advances in Hypercomplex Analysis`*, Springer.

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