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A function is called even if f(x) = f(-x)
A function is called odd if f(x) = -f(-x)
The logarithm function is defined for x values in the range 0 < x < ∞.
Radius of convergence: |x| < 1 and x = 1.
Logarithm Base 10 Function : y = log10(x) |
Logarithm Base 1/2 Function : y = log1/2(x) |
Logarithm Base 2 Function : y = log2(x) |
a = 1/2 | |
a = 2 | |
a = e | |
a = 10 |
This function can be used for the approximation of the prime counting function π(x).
This relation is also known as prime number theorem.
Formulars of the logistic function:
The logistic function is the solution of the logistic differential equation.
Using the substitution u = 1 + ex and u' = ex gives du = exdx.
The Bi are the Bernoulli Numbers.
The Bernoulli numbers Bi and Bernoulli numbers Bi* differ only in the value of i= 1. B1=-1/2 and B1* = 1/2, where they only have different sign. All the other values are the same.
i | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | ... |
Bi | 1 | ±1/2 | 1/6 | 0 | -1/30 | 0 | 1/42 | 0 | -1/30 | 0 | 5/66 | 0 | -291/2730 | ... |
y = x-4 |
y = x-3 |
y = x-2 |
y = x-1 |
y = x0 = 1 |
y = x1 = x |
y = x2 |
y = x3 |
y = x4 |
y = x5 |
y = x24 |
y = x25 |
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