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Integrating and differentiating piecewise functions with Maxima

Unit step function x# Φ Φs sign ≡ 1 x# sign + 2 / 12 / x# 2 x# abs * / + if 11 line :

4 appVersion 0.99.7822.147

f# x# int f# x# int Maxima :

if c1 greathar than c2 then a else b. c1# c2# a# b# Φ_GT a# c1# c2# - Φ * b# c1# c2# - Φ * - b# + 11 line :

Max of two expressions u# v# Φ_MAX u# v# u# v# Φ_GT 11 line :

Min of two expressions u# v# If_MIN v# u# u# v# Φ_GT 11 line :

if c equals zero then a else zero c# a# Φ_EQ a# c# Φε + Φ c# Φε - Φ - (* 11 line :

if c1 equals c2 then a else b c1# c2# a# b# Φ_EQ b# c1# c2# - a# Φ_EQ + c1# c2# - b# Φ_EQ - 11 line :

Φs sign :

Φε 0.000000001 :

h 0.0000001 :

t I t g t int 11 line :

t In t I t 0 ≡ at τ g τ 0 t int + 11 line :

Numerical and symbolic derivatives.

t D Φs abs : t g t diff 21 line :

t Dn t h + g t h - g - 2 h * / 11 line :

Inequalities

c1 t 2^2 t * - :

c2 3 :

a 0.5 t * 1 + :

b 2 t sin * :

t f c1 c2 > a b if 11 line :

t g c1 c2 a b Φ_GT 11 line :

x f 0.1 + x g 2 1 sys x In 0.1 + x I 2 1 sys x D 0.1 + x Dn 2 1 sys

Equalities

Φε Clear 1

x f 1 x 0 ≡ 2 x 1 ≡ 3 x 2 ≡ 0 cases :

Nuerically this have a poor representation and significance.

This other is better

x f 1 x 0 - abs Φε < 2 x 1 - abs Φε < 3 x 2 - abs Φε < 0 cases :

Same using unit step Φ

x g x 010 Φ_EQ x 120 Φ_EQ + x 230 Φ_EQ + 11 line :

Use a big epsilon for this example

Φε 0.01 :

x f 0.1 + x g 2 1 sys x In 0.1 + x I 2 1 sys x Dn 0.1 + x D 2 1 sys

Expected value for the integral is zero, but for derivatives is infinity at the discontinuities.

Max and Min

a t :

b t 2^:

t f a b 12 mat max 11 line :

t g a b Φ_MAX 11 line :

x f 0.1 + x g 2 1 sys x In 0.1 + x I 2 1 sys x Dn 0.1 + x D 2 1 sys

Alvaro