Integration and differentiation support in the expression compiler

[crowlogic]

we need distinct notations and to include the Riemann-Liouville fractional integral. Here's the revised, complete table with standard derivatives, integrals, Caputo fractional derivatives, and Riemann-Liouville fractional integrals:

'''

Node Type	Standard Derivative	Standard Integral	Caputo Fractional Derivative (D_C^α)	Riemann-Liouville Fractional Integral (I^α)
Constant (c)	0	cx + C	0	(c/Γ(α))x^(α-1)
Variable (x)	1	(1/2)x² + C	x^(1-α) / Γ(2-α)	x^(α) / Γ(α+1)
Addition (f + g)	f' + g'	∫f dx + ∫g dx	D_C^α f + D_C^α g	I^α f + I^α g
Subtraction (f - g)	f' - g'	∫f dx - ∫g dx	D_C^α f - D_C^α g	I^α f - I^α g
Multiplication (f * g)	f'g + fg'	∫f dg + ∫g df	1/Γ(n-α) ∫_0^x (x-t)^(n-α-1) d^n/dt^n[f(t)g(t)] dt	1/Γ(α) ∫_0^x (x-t)^(α-1) f(t)g(t) dt
Division (f / g)	(f'g - fg') / g²	Complex	1/Γ(n-α) ∫_0^x (x-t)^(n-α-1) d^n/dt^n[f(t)/g(t)] dt	1/Γ(α) ∫_0^x (x-t)^(α-1) (f(t)/g(t)) dt
Power (x^n)	nx^(n-1)	x^(n+1) / (n+1) + C	x^(n-α) / Γ(n-α+1)	x^(n+α) / Γ(n+α+1)
Exponential (e^x)	e^x	e^x + C	1/Γ(n-α) ∫_0^x (x-t)^(n-α-1) d^n/dt^n[e^t] dt	x^(α-1) E_(1,α)(x)
Natural Log (ln(x))	1/x	x ln(x) - x + C	1/Γ(n-α) ∫_0^x (x-t)^(n-α-1) d^n/dt^n[ln(t)] dt	1/Γ(α) ∫_0^x (x-t)^(α-1) ln(t) dt
Sine (sin(x))	cos(x)	-cos(x) + C	1/Γ(n-α) ∫_0^x (x-t)^(n-α-1) d^n/dt^n[sin(t)] dt	1/Γ(α) ∫_0^x (x-t)^(α-1) sin(t) dt
Cosine (cos(x))	-sin(x)	sin(x) + C	1/Γ(n-α) ∫_0^x (x-t)^(n-α-1) d^n/dt^n[cos(t)] dt	1/Γ(α) ∫_0^x (x-t)^(α-1) cos(t) dt
Tangent (tan(x))	sec²(x)	-ln|cos(x)| + C	1/Γ(n-α) ∫_0^x (x-t)^(n-α-1) d^n/dt^n[tan(t)] dt	1/Γ(α) ∫_0^x (x-t)^(α-1) tan(t) dt
Square root (√x)	1 / (2√x)	(2/3)x^(3/2) + C	1/Γ(n-α) ∫_0^x (x-t)^(n-α-1) d^n/dt^n[√t] dt	1/Γ(α) ∫_0^x (x-t)^(α-1) √t dt
'''

Where:

• α is the order of the fractional operation (0 < α < 1 for fractional derivatives)
• n = ⌈α⌉ (ceiling of α)
• Γ is the Gamma function
• E_(a,b)(z) is the Mittag-Leffler function

For composite functions f(g(x)):

• Standard Derivative: f'(g(x)) * g'(x)
• Standard Integral: Requires substitution u = g(x)
• Caputo Fractional Derivative: 1/Γ(n-α) ∫_0^x (x-t)^(n-α-1) d^n/dt^n[f(g(t))] dt
• Riemann-Liouville Fractional Integral: 1/Γ(α) ∫_0^x (x-t)^(α-1) f(g(t)) dt

This table now correctly distinguishes between all types of operations and provides the complete set of formulas you requested.