The duplication formula takes the form $${\displaystyle 2^{1-s}\operatorname {Li} _{s}(z^{2})=\operatorname {Li} _{s}(z)+\operatorname {Li} _{s}(-z).}$$ The general multiplication formula is in the form of a Gauss sum or discrete Fourier transform: $${\displaystyle k^{1-s}\operatorname {Li} _{s}(z^{k})=\sum … See more In mathematics, the multiplication theorem is a certain type of identity obeyed by many special functions related to the gamma function. For the explicit case of the gamma function, the identity is a product of values; … See more The polygamma function is the logarithmic derivative of the gamma function, and thus, the multiplication theorem becomes additive, instead of multiplicative: for $${\displaystyle m>1}$$, and, for See more The periodic zeta function is sometimes defined as $${\displaystyle F(s;q)=\sum _{m=1}^{\infty }{\frac {e^{2\pi imq}}{m^{s}}}=\operatorname {Li} _{s}\left(e^{2\pi iq}\right)}$$ where Lis(z) is the See more The multiplication theorem takes two common forms. In the first case, a finite number of terms are added or multiplied to give the relation. In the second case, an infinite number of … See more The duplication formula and the multiplication theorem for the gamma function are the prototypical examples. The duplication formula for the gamma function is See more For the Hurwitz zeta function generalizes the polygamma function to non-integer orders, and thus obeys a very similar multiplication theorem: See more The duplication formula for Kummer's function is and thus resembles … See more WebProblems on Gauss Law. Problem 1: A uniform electric field of magnitude E = 100 N/C exists in the space in the X-direction. Using the Gauss theorem calculate the flux of this field through a plane square area of edge 10 cm placed in the Y-Z plane. Take the normal along the positive X-axis to be positive.

Gauss

WebMar 24, 2024 · Gamma functions of argument can be expressed using a triplication formula (51) The general result is the Gauss multiplication formula (52) The gamma function is also related to the Riemann zeta … WebApr 5, 2010 · Technique 1: Pair Numbers. Pairing numbers is a common approach to this problem. Instead of writing all the numbers in a single column, let’s wrap the numbers … meme pass the buck

Legendre

Webhow the results of Euler and Gauss can be fully understood only in the context of class field theory. Finally, in order to bring class field theory down to earth, the book explores some of the magnificent formulas of complex multiplication. The central theme of the book is the story of which primes p can be expressed in the form x2 + ny2. WebFeb 16, 2024 · From the definition of the Gaussian distribution, X has probability density function : fX(x) = 1 σ√2πexp( − (x − μ)2 2σ2) From the definition of the expected value … WebNov 29, 2024 · 4.2 Duplication and Multiplication formula,41);$ ! duplication formula Theorem 6 (Legendre, 1809) ... Set x=1/n in the Gauss multiplication formula. meme pack for editing