rewrite

Rewrite expression in terms of another function

Syntax

rewrite(expr,target)

Description

rewrite(expr,target) rewrites the symbolic expression expr in terms of the target function target. The rewritten expression is mathematically equivalent to the original expression. If expr is a vector or matrix, rewrite acts element-wise on expr.

Examples

Rewrite Between Trigonometric and Exponential Functions

Rewrite any trigonometric function in terms of the exponential function by specifying the target 'exp'.

syms x
sin2exp = rewrite(sin(x), 'exp')
tan2exp = rewrite(tan(x), 'exp')

sin2exp =
(exp(-x*1i)*1i)/2 - (exp(x*1i)*1i)/2

tan2exp =
-(exp(x*2i)*1i - 1i)/(exp(x*2i) + 1)

Rewrite the exponential function in terms of any trigonometric function by specifying the trigonometric function as the target. For a full list of targets, see target.

syms x
exp2sin = rewrite(exp(x), 'sin')
exp2tan = rewrite(-(exp(x*2i)*1i - 1i)/(exp(x*2i) + 1), 'tan')

exp2sin =
1 - 2*sin((x*1i)/2)^2 - sin(x*1i)*1i
exp2tan =
-(((tan(x) - 1i)*1i)/(tan(x) + 1i) + 1i)/...
    ((tan(x) - 1i)/(tan(x) + 1i) - 1)

Simplify exp2tan into the expected form by using simplify.

exp2tan = simplify(exp2tan)

exp2tan =
tan(x)

Rewrite Between Trigonometric Functions

Rewrite any trigonometric function in terms of any other trigonometric function by specifying the target. For a full list of targets, see target.

Rewrite tan(x) in terms of the sine function by specifying the target 'sin'.

syms x
tan2sin = rewrite(tan(x), 'sin')

tan2sin =
-sin(x)/(2*sin(x/2)^2 - 1)

Rewrite Between Hyperbolic Functions and Trigonometric Functions

Rewrite any hyperbolic function in terms of any trigonometric function by specifying the trigonometric function as the target. For a full list of targets, see target.

Rewrite tanh(x) in terms of the sine function by specifying the target 'sin'.

syms x
tanh2sin = rewrite(tanh(x), 'sin')

tanh2sin =
(sin(x*1i)*1i)/(2*sin((x*1i)/2)^2 - 1)

Similarly, rewrite trigonometric functions in terms of hyperbolic functions by specifying the hyperbolic function as the target.

Rewrite Between Inverse Trigonometric Functions and Logarithm Function

Rewrite any inverse trigonometric function in terms of the logarithm function by specifying the target 'log'. For a full list of targets, see target.

Rewrite acos(x) and acot(x) in terms of the log function.

syms x
acos2log = rewrite(acos(x), 'log')
acot2log = rewrite(acot(x), 'log')

acos2log =
-log(x + (1 - x^2)^(1/2)*1i)*1i

acot2log =
(log(1 - 1i/x)*1i)/2 - (log(1i/x + 1)*1i)/2

Similarly, rewrite the logarithm function in terms of an inverse trigonometric function by specifying the inverse trigonometric function as the target.

Rewrite Elements of Matrix

Rewrite each element of a matrix by calling rewrite on the matrix.

Rewrite all elements of a matrix in terms of the exp function.

syms x
matrix = [sin(x) cos(x); sinh(x) cosh(x)];
rewrite(matrix, 'exp')

ans =
[ (exp(-x*1i)*1i)/2 - (exp(x*1i)*1i)/2, exp(-x*1i)/2 + exp(x*1i)/2]
[                 exp(x)/2 - exp(-x)/2,       exp(-x)/2 + exp(x)/2]

Rewrite Between Sine and Cosine Functions

Rewrite the cosine function in terms of the sine function. Here, rewrite replaces the cosine function using the identity cos(2*x) = 1 – 2*sin(x)^2 which is valid for any x.

syms x
rewrite(cos(x),'sin')

ans =
1 - 2*sin(x/2)^2

rewrite does not replace sin(x) with either $$-\sqrt{1-{\cos }^2\left(x\right)}$$ or $$\sqrt{1-{\cos }^2\left(x\right)}$$ because these expressions are not valid for all x. However, using the square of these expressions to replace sin(x)^2 is valid for all x. Thus, rewrite replaces sin(x)^2.

syms x
rewrite(sin(x),'cos')
rewrite(sin(x)^2,'cos')

ans =
sin(x)
ans =
1 - cos(x)^2

Input Arguments

expr — Input to rewrite

symbolic number | symbolic variable | symbolic expression | symbolic function | symbolic vector | symbolic matrix | symbolic multidimensional array

Input to rewrite, specified as a symbolic number, variable, expression, function, vector, matrix, or multidimensional array.

target — Target function

character vector

Target function, specified as a character vector. This table summarizes the rewriting rules for all allowed targets.

Target	Rewrites These Functions	In Terms of These Functions
'exp'	All trigonometric and hyperbolic functions including inverse functions	exp, log
'log'	All inverse trigonometric and hyperbolic functions	log
'sincos'	tan, cot, exp, sinh, cosh, tanh, coth	sin, cos
'sin', 'cos', 'tan', or 'cot'	sin, cos, exp, tan, cot, sinh, cosh, tanh, coth except the target	Target trigonometric function
'sinhcosh'	tan, cot, exp, sin, cos, tanh, coth	sinh, cosh
'sinh', 'cosh', 'tanh', 'coth'	tan, cot, exp, sin, cos, sinh, cosh, tanh, coth except the target	Target hyperbolic function
'asin', 'acos', 'atan', 'acot'	log, and all inverse trigonometric and inverse hyperbolic functions	Target inverse trigonometric function
'asinh', 'acosh', 'atanh', 'acoth'	log, and all inverse trigonometric and inverse hyperbolic functions	Target inverse hyperbolic function
'sqrt'	abs(x + 1i*y)	sqrt(x^2 + y^2)
'heaviside'	sign, triangularPulse, rectangularPulse	heaviside
'piecewise'	abs, heaviside, sign, triangularPulse, rectangularPulse	piecewise

Tips

• rewrite replaces symbolic function calls in expr with the target function only if the replacement is mathematically valid. Otherwise, it keeps the original function calls.