Editing Carl Friedrich Gauss (section)

=== Analysis ===
One of Gauss's first discoveries was the notion of the [[arithmetic-geometric mean]] (AGM) of two positive real numbers.{{sfn|Schlesinger|1933|p=41–57}} He discovered its relation to elliptic integrals in the years 1798–1799 through [[Landen's transformation]], and a diary entry recorded the discovery of the connection of [[Gauss's constant]] to [[lemniscatic elliptic functions]], a result that Gauss stated  "will surely open an entirely new field of analysis".<ref name="Cox">{{cite journal | last = Cox | first = David A. | author-link = David A. Cox | date = January 1984 | title = The Arithmetic-Geometric Mean of Gauss | url = https://www.researchgate.net/publication/248675540 | journal = [[L'Enseignement mathématique]] | volume = 30 | issue = 2 | pages = 275–330}}</ref> He also made early inroads into the more formal issues of the foundations of [[complex analysis]], and from a letter to Bessel in 1811 it is clear that he knew the "fundamental theorem of complex analysis" – [[Cauchy's integral theorem]] – and understood the notion of [[residue (complex analysis)|complex residues]] when integrating around [[pole (complex analysis)|poles]].<ref name="Stuhler" /><ref>Letter Gauss to Bessel from 18 December 1811, partly printed in the [https://gdz.sub.uni-goettingen.de/id/PPN236010751?tify=%7B%22pages%22%3A%5B96%5D%2C%22pan%22%3A%7B%22x%22%3A0.553%2C%22y%22%3A0.327%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.893%7D ''Collected Works'', Volume 8, pp. 90&ndash;92].</ref>

[[Pentagonal number theorem|Euler's pentagonal numbers theorem]], together with other researches on the AGM and lemniscatic functions, led him to plenty of results on [[Jacobi theta functions]],<ref name="Stuhler" /> culminating in the discovery in 1808 of the later called [[Jacobi triple product identity]], which includes Euler's theorem as a special case.<ref>{{cite book | last = Roy | first = Ranjan | author-link = Ranjan Roy | date = 2021 | edition = 2 | title = Series and Products in the Development of Mathematics | place = Cambridge | pages = 20–22 |volume=2 | publisher = Cambridge University Press | url = https://assets.cambridge.org/97811087/09453/frontmatter/9781108709453_frontmatter.pdf |isbn=9781108709378}}</ref> His works show that he knew modular transformations of order 3, 5, 7 for elliptic functions since 1808.{{sfn|Schlesinger|1933|pp=185-186}}{{efn|Later, these transformations were given by Legendre in 1824 (3th order), Jacobi in 1829 (5th order), [[Ludwig Adolf Sohncke|Sohncke]] in 1837 (7th and other orders).}}{{efn|In a letter to Bessel from 1828, Gauss commented: "Mr. Abel has [...] anticipated me, and relieves me of the effort [of publishing] in respect to one third of these matters ..."{{sfn|Schlesinger|1933|p=41}}}}

Several mathematical fragments in his [[Nachlass]] indicate that he knew parts of the modern theory of [[modular forms]].<ref name="Stuhler" /> In his work on the [[Multivalued function|multivalued]] AGM of two complex numbers, he discovered a deep connection between the infinitely many values of the AGM and its two "simplest values".<ref name="Cox" /> In his unpublished writings he recognized and made a sketch of the key concept of [[fundamental domain]] for the [[modular group]].{{sfn|Schlesinger|1933|pp=101–106}}<ref name="Houzel">{{cite book | last = Houzel | first = Christian | editor-last1 = Goldstrein | editor-first1 = Catherine | editor-last2 = Schappacher | editor-first2 = Norbert | editor-last3 = Schwermer | editor-first3 = Joachim | editor-link1 = Catherine Goldstein | editor-link2 = Norbert Schappacher | editor-link3 = Joachim Schwermer | title = The Shaping of Arithmetic after C. F. Gauss's Disquisitiones Arithmeticae | publisher = Springer | place = Berlin, Heidelberg, New York| date = 2007 | page = 293 | chapter = Elliptic Functions and Arithmetic | doi = 10.1007/978-3-540-34720-0 | isbn = 978-3-540-20441-1 | url = https://link.springer.com/book/10.1007/978-3-540-34720-0}}</ref> One of Gauss's sketches of this kind was a drawing of a [[tessellation]] of the [[unit disk]] by "equilateral" [[hyperbolic triangle]]s with all angles equal to <math>\pi/4</math>.<ref>Printed in the [https://gdz.sub.uni-goettingen.de/id/PPN236010751?tify=%7B%22pages%22%3A%5B110%5D%2C%22view%22%3A%22info%22%7D ''Collected Works'', Volume 8, p. 104].</ref>

An example of Gauss's insight in analysis is the cryptic remark that the principles of circle division by compass and straightedge can also be applied to the division of the [[lemniscate curve]], which inspired Abel's theorem on lemniscate division.{{efn|This remark appears at article 335 of chapter 7 of ''Disquisitiones Arithmeticae'' (1801).}} Another example is his publication "Summatio quarundam serierum singularium" (1811) on the determination of the sign of [[quadratic Gauss sums]], in which he solved the main problem by introducing [[Gaussian binomial coefficient|q-analogs of binomial coefficient]]s and manipulating them by several original identities that seem to stem from his work on elliptic function theory; however, Gauss cast his argument in a formal way that does not reveal its origin in elliptic function theory, and only the later work of mathematicians such as [[Carl Gustav Jacob Jacobi|Jacobi]] and [[Hermite]] has exposed the crux of his argument.{{sfn|Schlesinger|1933|pp=122&ndash;123}}

In the "Disquisitiones generales circa series infinitam..." (1813), he provides the first systematic treatment of the general [[hypergeometric function]] <math>F(\alpha,\beta,\gamma,x)</math>, and shows that many of the functions known at the time are special cases of the hypergeometric function.{{sfn|Schlesinger|1933|pp=136–142}} This work is the first exact inquiry into [[Convergent series|convergence]] of infinite series in the history of mathematics.{{sfn|Schlesinger|1933|p=142}} Furthermore, it deals with infinite [[continued fraction]]s arising as ratios of hypergeometric functions, which are now called [[Gauss continued fraction]]s.{{sfn|Schlesinger|1933|pp=136–154}}

In 1823, Gauss won the prize of the Danish Society with an essay on [[conformal mapping]]s, which contains several developments that pertain to the field of complex analysis.{{sfn|Stäckel|1917|pp=90&ndash;91}} Gauss stated that angle-preserving mappings in the complex plane must be complex [[analytic function]]s, and used the later-named [[Beltrami equation]] to prove the existence of [[isothermal coordinates]] on analytic surfaces. The essay concludes with examples of conformal mappings into a sphere and an [[ellipsoid of revolution]].{{sfn|Bühler|1981|p=103}}

==== Numerical analysis ====
Gauss often deduced theorems [[Inductive reasoning|inductively]] from numerical data he had collected empirically.{{sfn|Schlesinger|1933|p=18}} As such, the use of efficient algorithms to facilitate calculations was vital to his research, and he made many contributions to [[numerical analysis]], such as the method of [[Gaussian quadrature]], published in 1816.<ref>{{cite book | last = Gautschi | first = Walter | author-link = Walter Gautschi | edition = 1 | title = E.B. Christoffel. The Influence of his Work on Mathematics and the Physical Science | editor-last1 = Butzer | editor-first1 = Paul B. | editor-last2 = Fehér | editor-first2 = Franziska | editor-link1 = Paul Butzer | chapter = A Survey of Gauss-Christoffel Quadrature Formulae | place = Birkhäuser, Basel | year = 1981 | pages = 72–147 | publisher = Springer | doi = 10.1007/978-3-0348-5452-8_6 | isbn = 978-3-0348-5452-8 | chapter-url = https://link.springer.com/chapter/10.1007/978-3-0348-5452-8_6}}</ref>

In a private letter to [[Christian Ludwig Gerling|Gerling]] from 1823,<ref>{{Cite book|url=https://gdz.sub.uni-goettingen.de/id/PPN335994989?tify=%7B%22pages%22:%5B316%5D,%22view%22:%22info%22%7D|title=Briefwechsel zwischen Carl Friedrich Gauss und Christian Ludwig Gerling|publisher=Elsner}}</ref> he described a solution of a 4x4 system of linear equations with the [[Gauss-Seidel method]] – an "indirect" [[iterative method]] for the solution of linear systems, and recommended it over the usual method of "direct elimination" for systems of more than two equations.<ref>{{cite arXiv | author = Yousef Saad | author-link = Yousef Saad | title = Iterative Methods for Linear Systems of Equations: A Brief Historical Journey | date = 2 August 2019 | class = math.HO | eprint = 1908.01083v1}}</ref>

Gauss invented an algorithm for calculating what is now called [[discrete Fourier transform]]s when calculating the orbits of Pallas and Juno in 1805, 160 years before [[James Cooley|Cooley]] and [[John Tukey|Tukey]] found their similar [[Cooley-Tukey FFT algorithm|Cooley–Tukey algorithm]].<ref>{{cite journal | last1 = Cooley | first1 = James W. | first2 = John W. | last2 = Tukey | title = An algorithm for the machine calculation of complex Fourier series | journal = [[Mathematics of Computation]] | volume = 19 | issue = 90 | pages = 297–301 | year = 1965 | doi = 10.2307/2003354 | jstor = 2003354 | doi-access=free }}</ref> He developed it as a [[trigonometric interpolation]] method, but the paper ''Theoria Interpolationis Methodo Nova Tractata'' was published only posthumously in 1876,<ref>{{cite book | last = Gauss | first = C.F. | date = 1876 | title = Theoria Interpolationis Methodo Nova Tractata | place = Göttingen | language = la | pages = 265–327 | url = https://gdz.sub.uni-goettingen.de/id/PPN235999628?tify=%7B%22pages%22%3A%5B273%5D%2C%22pan%22%3A%7B%22x%22%3A0.524%2C%22y%22%3A0.333%7D%2C%22view%22%3A%22info%22%2C%22zoom%22%3A0.856%7D | publisher = K. Gesellschaft der Wissenschaften zu Göttingen}}</ref> well after [[Joseph Fourier]]'s introduction of the subject in 1807.<ref>{{cite journal | last1 = Heideman | first1 = Michael T. | last2 = Johnson| first2 = Don H. | last3 = Burrus | first3 = C. Sidney | author-link3 = C. Sidney Burrus | title = Gauss and the history of the fast Fourier transform | journal = IEEE ASSP Magazine | year = 1984 | volume = 1 | issue = 4 | pages = 14–21 | doi = 10.1109/MASSP.1984.1162257|s2cid=10032502 | url = http://www.cis.rit.edu/class/simg716/Gauss_History_FFT.pdf | archive-url=https://web.archive.org/web/20130319053449/http://www.cis.rit.edu/class/simg716/Gauss_History_FFT.pdf | archive-date = 19 March 2013 | url-status=live}}</ref>