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Analysis

One of Gauss's first discoveries was the notion of the arithmetic-geometric mean (AGM) of two positive real numbers. He discovered its relation to elliptic integrals in the years 1798–1799 through the 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 that "will surely open an entirely new field of analysis". 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 complex residues when integrating around poles.

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, culminating in the discovery in 1808 of the later called Jacobi triple product identity, which includes Euler's theorem as a special case. His works show that he knew modular transformations of order 3, 5, 7 for elliptic functions since 1808.

Several mathematical fragments in his Nachlass indicate that he knew parts of the modern theory of modular forms. In his work on the multivalued AGM of two complex numbers, he discovered a deep connection between the infinitely many values of the AGM to its two "simplest values". In his unpublished writings he recognized and made a sketch of the key concept of fundamental domain for the modular group. One of Gauss's sketches of this kind was a drawing of a tessellation of the unit disk by "equilateral" hyperbolic triangles with all angles equal to π / 4 {\displaystyle \pi /4}.

An example of Gauss's insight in the fields of 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. Another example is his publication "Summatio quarundam serierum singularium" (1811) on the determination of the sign of quadratic Gauss sum, in which he solved the main problem by introducing q-analogs of binomial coefficients and manipulating them by several original identities that seem to stem out of his work on elliptic functions theory; however, Gauss cast his argument in a formal way that does not reveal its origin in elliptic functions theory, and only the later work of mathematicians such as Jacobi and Hermite has exposed the crux of his argument.

In the "Disquisitiones generales circa series infinitam..." (1813), he provides the first systematic treatment of the general hypergeometric function F (α, β, γ, x) {\displaystyle F(\alpha,\beta,\gamma,x)}, and shows that many of the functions known at the time are special cases of the hypergeometric function. This work is the first one with an exact inquiry of convergence of infinite series in the history of mathematics. Furthermore, it deals with infinite continued fractions arising as ratios of hypergeometric functions which are now called Gauss continued fractions.

In 1823, Gauss won the prize of the Danish Society with an essay on conformal mappings, which contains several developments that pertain to the field of complex analysis. Gauss stated that angle-preserving mappings in the complex plane must be complex analytic functions, and used the later called 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.