Napier's constant is widely known as having an especial value as like as Euler's constant.
Napier's constant performs the main role expressing linear natural phenomena.
The reason stands on the characteristics that the derivative of exponential function is the exponential one itself.
2. Original differential equation and its solution
The original simple differential equation which is composed of two continuous functions of
and in all range of
is assumed to be given as in the following;
3. Algebraic and analytic expressions of exponential function
The special case of in Eq.(5) will be treated in the foregoing description.
According to such a treatment, Eq.(5) must coincide to Eq.(3).
So, the following equation can be described.
Although we have obtained the Eq.(7) which connects the algebraic and the analytic expressions of exponential function, the calculation process mentioned above is hardly seen. However, this process will find a side of the essence of the exponential function.
As the Napier's, the Euler's and pi's constants are given by ,
then, the following equation is calculated.
Although the three constants is thought to have no relation each other,
the value of the equation closely coincides to a unit by 0.11 percent error.
[Reference] Another relation has been found as below.
But, the value of this equation have the error of 0.98 percent more than the above one.
[The other approximation]
Although the following equation is also made by a numerical value adjustment,
the value of the equation closely coincides to the Napier's natural constant by 0.037 percent error.