Du Bois Reymond Constant

The constants Du Bois Reymond, (Paul David Gustav) Cn{displaystyle C_{n}} are defined by

Cn≡ ≡ ∫ ∫ 0∞ ∞ 日本語ddt(without tt)n日本語dt− − 1{displaystyle C_{n}equiv int _{0}^{infty }{leftY}{{d over dt}left({sin t over t}right)^{n}}}{right

These constants can also be written as:

Cn=2␡ ␡ k=1∞ ∞ (1+xk2)(− − n/2){displaystyle C_{n}=2sum _{k=1}^{infty }(1+x_{k}^{2})^{(-n/2)}}}}

where xk{displaystyle x_{k}} is the root k-sima

t=So... (t){displaystyle t=tan(t)}

In addition we have the following series

␡ ␡ n=1∞ ∞ 1xk2=110{displaystyle sum _{n=1}^{infty }{1 over x_{k}{2}{2}}={1 over 10}}}

The following graph shows the representation of the function

日本語ddt(without tt)n日本語{displaystyle leftATA{d over dt}left({sin t over tright}^{n}right

for the first four values n{displaystyle n}

The numerical integration of this function is difficult. The first four values of these constants are:

C1{displaystyle C_{1}} dive

C2≈ ≈ 0.1945{displaystyle C_{2}approx 0.1945}

C3≈ ≈ 0.028254{displaystyle C_{3approx 0.028254}

C4≈ ≈ 0.00524054{displaystyle C_{4}approx 0.00524054}

The constant pairs of Bois Reymond can be calculated analytically as polynomials in e2{displaystyle e^{2}}.

C2=12(e2− − 7){displaystyle C_{2}={1 over 2}(e^{2}-7)}

C4=18(e4− − 4e2− − 25){displaystyle C_{4}={1 over 8}(e^{4}-4e^{2-25}}}}