MAXIMA

MAXIMA Quick Reference

Fourier Transform

Fourier Series Expansion

			Function: totalfourier (f, x, p)
				Returns fourexpand (foursimp (fourier (f, x, p)), x, p, 'inf).
			Function: opsubst (f, g, e)
    				The function opsubst is similar to the function subst, except that opsubst only makes substitutions for the operators in an expression. In general, when f is an operator in the expression e, substitute g for f in the expression e.
    				To determine the operator, opsubst sets inflag to true. This means opsubst substitutes for the internal, not the displayed, operator in the expression.
    				To use this function write first load("opsubst").
			Example:
			     (%i) load("fourie")$
			     (%i) f(x):=x;
			     (%i) totalfourier(f(x),x,1/2);
			     (%o) -('sum((-1)^n*sin(2*%pi*n*x)/n,n,1,inf))/%pi
			     (%i) FF:subst(3,inf,%);
			     (%o) -('sum((-1)^n*sin(2*%pi*n*x)/n,n,1,3))/%pi
			     (%i) load("opsubst")$
			     (%i) FG:opsubst(sum,'sum,FF);
			     (%o) -sum((-1)^n*sin(2*%pi*n*x)/n,n,1,3)/%pi
			     (%i) define(g(x),FG);
			     (%o) g(x):=-sum((-1)^n*sin(2*%pi*n*x)/n,n,1,3)/%pi

Laplace Tranceform

LAPLACE ... Laplace Tranceform

Function: laplace (expr, t, s)
 The Laplace transform of expr with respect to the variable t and transform parameter s. 
Example:
 Delta Function
	(%i) laplace(delta(t), t, s);
	(%o) 1 
 Unit Step Function
    (%i) laplace(1, t, s);
    (%o) 1/s 
 Trigonometric Function
	(%i) laplace(sin(t), t, s);
	(%o) 1/(s^2+1)
	(%i) laplace(cos(t), t, s);
	(%o) s/(s^2+1) 
 Exponential Function
	(%i) laplace(%e^(-a*t),t,s);
	(%o) 1/(s+a)
	(%i) laplace(%e^(-a*t)*t^2, t, s);
	(%o) 2/(s+a)^3

ILT ... Inverse Laplace Tranceform

Function: ilt (expr, t, s)
    The inverse Laplace transform of expr with respect to t and parameter s 
Example:
    (%i) ilt(1/s^2, s, t);
    (%o) t
    (%i) ilt(1/s^3, s, t);
    (%o) t^2/2

    (%i) ilt(1/(s+a), s, t);
    (%o) %e^-(a*t)
    (%i) ilt(1/(s+a)^2, s, t);
    (%o) t*%e^-(a*t)

    (%i) ilt((s+a)/((s+a)^2+w^2), s, t); 
    (%i) ilt(1/((s+2)^2*(s+3)), s, t); 
    (%i) ilt(1/(s*((s+2)^2+1)), s, t);

Taylor Expansion

maxima_taylor

			Function: taylor (expr, x, a, n)
			Function: taylor (expr, [x_1, x_2, ...], a, n)
			Function: taylor (expr, [x, a, n, 'asymp])
			Function: taylor (expr, [x_1, x_2, ...], [a_1, a_2, ...], [n_1, n_2, ...])
			Function: taylor (expr, [x_1, a_1, n_1], [x_2, a_2, n_2], ...)

			    Expands of the expression expr in a truncated Taylor or Laurent series in the variable x around the point a, containing terms through (x - a)^n. 

			Example:

			Trigonometric Expansion
			(%i) taylor(sin(x), x, 0, 7);
			(%o) x-x^3/6+x^5/120-x^7/5040
			(%i) taylor(cos(x), x, 0, 7);
			(%o) 1-x^2/2+x^4/24-x^6/720 Exponential Function
			(%i) taylor(exp(x), x, 0, 7 );
			(%o) 1+x+x^2/2+x^3/6+x^4/24+x^5/120+x^6/720+x^7/5040 Summation of Geometric Progression
			(%i) taylor(1/(1-x), x, 0, 7 );
			(%o) 1+x+x^2+x^3+x^4+x^5+x^6+x^7
			(%i) taylor(1/(1+x), x, 0, 7 );
        		(%o) 1-x+x^2-x^3+x^4-x^5+x^6-x^7 
        		(%i) taylor(log(1-x),x,0,7);
        		(%o) -x-x^2/2-x^3/3-x^4/4-x^5/5-x^6/6-x^7/7

Trigonometric

TRIGEXPAND ... Trigonometric Expansion

Function: trigexpand (expr)
	Expands trigonometric and hyperbolic functions of sums of angles and of multiple angles occurring in expr. 
Example:
Trigonometric expansion
	(%i) trigexpand(sin(x+y));
	(%o) sin(x)*cos(y)+cos(x)*sin(y)
	(%i) trigexpand(cos(x+y));
	(%o) cos(x)*cos(y)-sin(x)*sin(y)
	(%i) trigexpand(tan(x+y));
	(%o) (tan(y)+tan(x))/(1-tan(x)*tan(y))
Double-Angle Formula
	(%i) trigexpand(sin(2*x));
	(%o) 2*cos(x)*sin(x)
	(%i) trigexpand(cos(2*x));
	(%o) cos(x)^2-sin(x)^2
	(%i) trigexpand(tan(2*x));
	(%o) 2*tan(x)/(1-tan(x)^2)

TRIGREDUCE ... Combines products and powers of trigonometric and hyperbolic

Function:trigreduce(expr, x) 
Function:trigreduce(expr)
	Combines products and powers of trigonometric and hyperbolic
Example:
	(%i) trigreduce(sin(x)^2);
	(%o) (1-cos(2*x))/2
	(%i) trigreduce(sin(x)^3);
	(%o) (3*sin(x)-sin(3*x))/4
	(%i) trigreduce(cos(x)^2);
	(%o) (cos(2*x)+1)/2
	(%i) trigreduce(cos(x)^3);
	(%o) (cos(3*x)+3*cos(x))/4
	(%i) trigreduce(-sin(x)^2+3*cos(x)^2+x);
	(%o) cos(2*x)/2+3*(cos(2*x)/2+1/2)+x-1/2

Matrix

DETERMINANT ... Determinant of a Matrix

Function: determinant (M)
    Returns the determinant of the matrix M. 
Example:
    (%i) determinant( matrix( [1,2], [2,1] ) );
    (%o) -3
    (%i) determinant( matrix( [a,b], [c,d] ) );
    (%o) a*d-b*c
    (%i) determinant( matrix( [a,b,c],[d,e,f],[g,h,i] ) );
    (%o) a*(e*i-f*h)-b*(d*i-f*g)+c*(d*h-e*g)