3-Algebra-Function-Kinds-Exponential

error function

Functions {error function} can have value (2 / (pi^0.5)) * integral from t = 0 to t = z of e^(-t^2) * dt.

exponential function

Functions {exponential function, power} can have constant base {radix, base} {base, radix} raised to variable power: 10^x and e^x. exp(x) equals e^x if base is e or equals 10^x if base is 10. Exponential functions are logarithmic-function inverses.

Laplace transform

Functions {Laplace transform} can be integral from t = 0 to t = +infinity of e^(-s*t) * F(t) * dt. If function value is 1/s, then F(t) = 1. Integral from x = 0 to x = infinity of integral of y = 0 to y = infinity of e^(-u*x - v*y) * F(x,y) * dx * dy is Laplace transform.

logarithmic function

Functions {logarithmic function, exponential} can use variable powers of constant bases. ln(x) depends on base e {natural logarithm, base}. log(x) depends on base 10 {common logarithm}. Logarithms are exponential inverses.

Taking logarithm of value gives exponent to use with base. log(100) = x = log(10^x) = 2. ln(e^x) = x.

product

Product logarithm equals sum of factor logarithms {law of exponents}: log(M * N) = log(M) + log(N).

Exponential-function product equals exponential function of exponent sum: exp(x) * exp(y) = exp(x + y).

hyperbola

Logarithm equals area under hyperbola integrated from 1 to y, because hyperbola has equation y = 1/x, and integral of 1/y is logarithm.

line

Exponential equals 1 + n + n^2 / 2, for value n. Therefore, exponential equals 1 plus value plus area under line plus area under line at 45-degree angle.

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Date Modified: 2022.0225