DMelt:JMathlab/8 Symbolic Transformations

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Symbolic Transformations

Substitution

Parts of an expression may be replaced by other expressions using subst(a,b,c): a is substituted for b in c. This is a powerful function with many uses.

First, it may be used to insert numbers for variables, in the example $3$ for $x$ in der formula $2\sqrt{x}\cdot e^{-x^2}$.

>> syms x
>> a=2*sqrt(x)*exp(-x^2);
>> subst(3,x,a)             
ans = 4.275E-4

Second, one can replace a symbolic variable by a complex term. The expression is automatically updated to the canonical format. In the following example $z^3+2$ is inserted for $x$ in $x^3+2x^2+x+7$.

>> syms x,z
>> p=x^3+2*x^2+x+7;
>> subst(z^3+2,x,p)         
ans = z^9+8*z^6+21*z^3+25

Finally, the term b itself may be a complex expression (in the example $z^2+1$). Jasymca then tries to identify this expression in c (example: $\frac{z\cdot x^3}{\sqrt{z^2+1}}$). This is accomplished by solving the equation $a = b$ for the symbolic variable in b (example: $z$), and inserting the solution in c. This does not always succeed, or there may be several solutions, which are returned as a vector.

>> syms x,y,z
>> c=x^3*z/sqrt(z^2+1);
>> d=subst(y,z^2+1,c)
d = [ x^3*sqrt(y-1)/sqrt(sqrt(y-1)^2+1)  
     -x^3*sqrt(y-1)/sqrt(sqrt(y-1)^2+1) ]
>> d=trigrat(d)
d = [ x^3*sqrt(y-1)/sqrt(y)  
     -x^3*sqrt(y-1)/sqrt(y) ]

Simplifying and Collecting Expressions

The function trigrat(expression) applies a series of algorithms to expression.

All numbers are transformed to exact format. Trigonometric functions are expanded to complex exponentials. Addition theorems for the exponentials are applied. Square roots are calculated and collected. Complex exponentials are backtransformed to trigonometric functions.

It is often required to apply float(expression) to the final result.

>> syms x
>> trigrat(sin(x)^2+cos(x)^2)
ans = 1
>> b=sin(x)^2+sin(x+2*pi/3)^2+sin(x+4*pi/3)^2;
>> trigrat(b)
ans = 3/2
>> trigrat(i/2*log(x+i*pi))
ans = 1/4*i*log(x^2+pi^2)+(1/2*atan(x/pi)-1/4*pi)
>> trigrat(sin((x+y)/2)*cos((x-y)/2))
ans = 1/2*sin(y)+1/2*sin(x)
>> trigrat(sqrt(4*y^2+4*x*y-4*y+x^2-2*x+1))
ans = y+(1/2*x-1/2)

trigexpand(expression) expands trigonometric expressions to complex exponentials. It is the first step of the function trigrat above.

>> syms x
>> trigexp(i*tan(i*x))
ans = (-exp(2*x)+1)/(exp(2*x)+1)
>> trigexp(atan(1-x^2))
ans = -1/2*i*log((-x^2+(1-1*i))/(x^2+(-1-1*i)))
More information on this topic is in DMelt books