MAXIMA Quick Reference

Fourier Transform

      Fourier Series Expansion

        • maxima_totalfourier
          			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").
          			     (%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. 
           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 
              (%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. 
            			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 


            TRIGEXPAND ... Trigonometric Expansion

            Function: trigexpand (expr)
            	Expands trigonometric and hyperbolic functions of sums of angles and of multiple angles occurring in expr. 
            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) 
            	Combines products and powers of trigonometric and hyperbolic
            	(%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


            DETERMINANT ... Determinant of a Matrix

            Function: determinant (M)
                Returns the determinant of the matrix M. 
                (%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) 


            Bode Diagram