伯努利功率系列/伯努利反功率系列软件

函数将普通幂级数转换为加权伯努利多项式的幂级数，反之亦然。

软件应用简介

Forward: Function input c is the ordinary coefficient vector, real-valued . Function output d is the Bernoulli coefficient vector, real-valued. Use row vectors.

d = bernoulli_power_series(c)

Inverse: Function output c is the ordinary coefficient vector, real-valued . Function input d is the Bernoulli coefficient vector, real-valued. Use row vectors.

c = inverse_bernoulli_power_series(d)

Thorough theory can be found here. Methodology relies on definition of Bernoulli polynomials via inverted Dirichlet series.

https://qr.ae/pNvLNt

Quick explanation

Define an ordinary power series, where c_k | k=0,1,2… denotes the ordinary coefficient vector

y(x) = c_0 + xc_1 + x^2c_2…

Define a Bernoulli power series , where d_k | k=0,1,2… denotes the Bernoulli coefficient vector

y(x) = B_0(x)d_0 + B_1(x)d_1 + B_2(x)d_2…

Forward transform:

Function input is the finite ordinary coefficient vector c_k | k=0,1,2…K and function output is equal-length Bernoulli coefficient vector d_k | k=0,1,2…K

Inverse transform:

Function output is the finite ordinary coefficient vector c_k | k=0,1,2…K and function input is equal-length Bernoulli coefficient vector d_k | k=0,1,2…K

界面展示

结果示意

规格 价