# Réponse finale obtenue du bot:
response = « A fascinating topic!
In this response, I’ll provide an overview of the properties of functions used in differential and integral calculus, series and sequences, complex analysis, and real analysis.
**Calculus**
* **Differential Calculus**: Properties of derivatives:
+ Linearity: `(f + g)’ = f’ + g’`
+ Product Rule: `(fg)’ = f’g + fg’`
+ Chain Rule: `(f ∘ g)’ = (f’ ∘ g) \* g’`
+ Power Rule: `((x^n))’ = n x^(n-1)`
* **Integral Calculus**: Properties of integrals:
+ Linearity: `∫(af + bg) dx = a ∫f dx + b ∫g dx`
+ Integration by Parts: `∫f(x) g'(x) dx = f(x) g(x) | ∫ – ∫g(x) f'(x) dx`
**Series and Sequences**
* **Convergence Tests**: For infinite series:
+ Nth Term Test: If the nth term of a series converges to 0, then the series converges.
+ Ratio Test: If the ratio of consecutive terms approaches 0 or ∞, then the series converges.
+ Root Test: If the root of the absolute value of the terms approaches 1 or less than 1, then the series converges.
* **Properties of Series**:
+ Linearity: `∑(af + bg) = a ∑f + b ∑g`
+ Absolute Convergence: A series with absolute values converging also converges.
**Complex Analysis**
* **Properties of Complex Functions**: For functions of the form f(z) = u(x,y) + iv(x,y):
+ Linearity: `(af + bg)’ = a f’ + b g’`
+ Cauchy-Riemann Equations: `∂u/∂x = ∂v/∂y` and `∂u/∂y = -∂v/∂x`
* **Cauchy’s Integral Formula**: For functions analytic on a simply connected domain:
+ `f(z) = (1/(2πi)) ∫_C ((f(w)/(w-z))) dw`
**Real Analysis**
* **Properties of Real Functions**: For functions defined on intervals [a,b]:
+ Linearity: `(af + bg)’ = a f’ + b g’`
+ Continuity: A function is continuous at a point if its limit exists.
+ Differentiability: A function is differentiable at a point if its derivative exists.
These are just some of the key properties in each area. If you’d like me to elaborate or provide more examples, feel free to ask! »