In mathematics, the Mercator series or Newton–Mercator series is the Taylor series for the natural logarithm: ln ⁡ ( 1 + x ) = x − x 2 2 + x 3 3 − x 4...

The transverse Mercator map projection (TM, TMP) is an adaptation of the standard Mercator projection. The transverse version is widely used in national...

{x^{3}}{3}}-\cdots .\end{aligned}}} The last series is known as Mercator series, named after Nicholas Mercator (since it was published in his 1668 treatise...

Mercator may refer to: Marius Mercator (c. 390–451), a Catholic ecclesiastical writer Arnold Mercator, a 16th-century cartographer Gerardus Mercator,...

Mercator is a series of combine harvesters produced by the German agricultural company Claas in Harsewinkel. Initially called Senator, the Mercator series...

Nicholas (Nikolaus) Mercator (c. 1620, Holstein – 1687, Versailles), also known by his German name Kauffmann, was a 17th-century mathematician. He was...

The Universal Transverse Mercator (UTM) is a map projection system for assigning coordinates to locations on the surface of the Earth. Like the traditional...

{\pi }{4}}} , but is not absolutely convergent. The Mercator series provides an analytic power series expression of the natural logarithm, given by ∑ n...

Rumold Mercator (Leuven, 1541 – Duisburg, 31 December 1599) was a cartographer, son of Gerardus Mercator and brother of Arnold Mercator, both also cartographers...

The Martin P4M Mercator was a maritime reconnaissance aircraft built by the Glenn L. Martin Company. The Mercator was an unsuccessful contender for a United...

modern concept used in computing. Matrix logarithm Mel scale Mercator projection Mercator series Moment magnitude scale John Napier Napierian logarithm Natural...

\end{aligned}}} This is the Taylor series for ln ⁡ x {\displaystyle \ln x} around 1. A change of variables yields the Mercator series: ln ⁡ ( 1 + x ) = ∑ k = 1...

their own work on infinitesimal calculus. History of group theory Mercator series Tangent Wikimedia Commons has media related to Johannes Hudde. J.P...

Weisstein, Eric W. "Mercator Series". MathWorld--A Wolfram Web Resource. Retrieved 2024-04-24. To extend the utility of the Mercator series beyond its conventional...

important arbitrary dimension n identity can be obtained from the Mercator series expansion of the logarithm when the expansion converges. If every eigenvalue...

The Bowring series of the transverse mercator published in 1989 by Bernard Russel Bowring gave formulas for the Transverse Mercator that are simpler to...

whose logarithm one attempts to take is invertible.) Then, by the Mercator series log ⁡ ( 1 + x ) = x − x 2 2 + x 3 3 − x 4 4 + ⋯ {\displaystyle \log(1+x)=x-{\frac...

Transverse Mercator projection has many implementations. Louis Krüger in 1912 developed one of his two implementations that expressed as a power series in the...

it as a superior alternative to the commonly used Mercator projection, on the basis that the Mercator projection greatly distorts the relative sizes of...

negative real numbers from the complex plane. Another example: The Mercator series ln ⁡ ( 1 + u ) = ∑ n = 1 ∞ ( − 1 ) n + 1 n u n = u − u 2 2 + u 3 3...

^{n}\exp(\operatorname {tr} (\log(I_{n}-A/\lambda ))).} Hence, by virtue of the Mercator series, p ( λ ) = λ n exp ⁡ ( − tr ⁡ ∑ m = 1 ∞ ( A λ ) m m ) , {\displaystyle...

is, expanding in terms of the non-negative matrix 1−ρ in the formal Mercator series for the logarithm, − ⟨ ln ⁡ ρ ⟩ = ⟨ 1 − ρ ⟩ + 1 2 ⟨ ( 1 − ρ ) 2 ⟩ +...

convergence, an alternating series called the Mercator series expresses the logarithm function over the interval (0,2). Since the series is negative in (0,1)...

reputation of the work of Gerard Mercator, and for his portraits of Francis Drake. He inherited and republished the plates of Mercator, thus reviving his legacy...

Proceedings of the Royal Society. By way of illustration he gives the Mercator series for the logarithm (denoted l.) without mention of radius of convergence:...

=e^{D}-1} where D is differentiation with respect to x, we have, from the Mercator series, D e D − 1 = log ⁡ ( Δ + 1 ) Δ = ∑ n = 0 ∞ ( − Δ ) n n + 1 . {\displaystyle...

earthquake Gerard Mercator, Flemish cartographer – Mercator projection. Nicholas Mercator, German mathematician – Mercator series. John Mercer, British...

Saint-Vincent - Jesuit mathematician who independently discovered the Mercator series, the expansion of log (1 + x) in ascending powers of x Saint Francis...

onto a plane.[citation needed] The most well-known map projection is the Mercator projection.: 45  This map projection has the property of being conformal...

{ad} _{Z}\|<\log 2~~,\end{aligned}}} the last step by virtue of the Mercator series expansion, it follows that and, thus, integrating, Z ( 1 ) = ∫ 0 1...