ZTrans Functions

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ZTrans Functions Support

The Fractal Science Kit fractal generator ZTrans functions are a set of complex transformations, their 1st and 2nd derivatives, and their inverses.

Base Functions:

ZTrans1(z) = 1/z
ZTrans2(z) = 1/(1+z)
ZTrans3(z) = 1/(1-z)
ZTrans4(z) = z/(1+z)
ZTrans5(z) = z/(1-z)
ZTrans6(z) = (1+z)/z
ZTrans7(z) = (1-z)/z
ZTrans8(z) = (z+1)/(z-1)
ZTrans9(z) = (z+1)/(1-z)
ZTrans10(z) = (z-1)/(z+1)
ZTrans11(z) = (1-z)/(z+1)
ZTrans12(z) = z + 1/z
ZTrans13(z) = z - 1/z
ZTrans14(z) = z + 1/(1+z)
ZTrans15(z) = z - 1/(1+z)
ZTrans16(z) = z + 1/(1-z)
ZTrans17(z) = z - 1/(1-z)
ZTrans18(z) = z + z/(1+z)
ZTrans19(z) = z - z/(1+z)
ZTrans20(z) = z + z/(1-z)
ZTrans21(z) = z - z/(1-z)
ZTrans22(z) = 1/(1+z) + 1/(1-z)
ZTrans23(z) = 1/(1+z) - 1/(1-z)
ZTrans24(z) = z/(1+z) + 1/(1-z)
ZTrans25(z) = z/(1+z) - 1/(1-z)
ZTrans26(z) = 1/(1+z) + z/(1-z)
ZTrans27(z) = 1/(1+z) - z/(1-z)
ZTrans28(z) = z/(1+z) + z/(1-z)
ZTrans29(z) = z/(1+z) - z/(1-z)
ZTrans30(z) = z^2/(1+z)
ZTrans31(z) = z^2/(1-z)
ZTrans32(z) = (1+z)/z^2
ZTrans33(z) = (1-z)/z^2
ZTrans34(z) = z^2 + 1/z
ZTrans35(z) = z^2 - 1/z
ZTrans36(z) = z^2 + 1/(1+z)
ZTrans37(z) = z^2 - 1/(1+z)
ZTrans38(z) = z^2 + 1/(1-z)
ZTrans39(z) = z^2 - 1/(1-z)
ZTrans40(z) = z^2 + z/(1+z)
ZTrans41(z) = z^2 - z/(1+z)
ZTrans42(z) = z^2 + z/(1-z)
ZTrans43(z) = z^2 - z/(1-z)
ZTrans44(z) = z*(1+z)
ZTrans45(z) = z*(1-z)
ZTrans46(z) = z*(z-1)
ZTrans47(z) = z^2 + 1
ZTrans48(z) = z^2 - 1
ZTrans49(z) = 1 - z^2
ZTrans50(z) = -1 - z^2

1st Derivatives:

ZTrans1Prime(z) = -1/z^2
ZTrans2Prime(z) = -1/(1+z)^2
ZTrans3Prime(z) = 1/(1-z)^2
ZTrans4Prime(z) = 1/(1+z)^2
ZTrans5Prime(z) = 1/(1-z)^2
ZTrans6Prime(z) = -1/z^2
ZTrans7Prime(z) = -1/z^2
ZTrans8Prime(z) = -2/(z-1)^2
ZTrans9Prime(z) = 2/(z-1)^2
ZTrans10Prime(z) = 2/(z+1)^2
ZTrans11Prime(z) = -2/(z+1)^2
ZTrans12Prime(z) = 1 - 1/z^2
ZTrans13Prime(z) = 1 + 1/z^2
ZTrans14Prime(z) = 1 - 1/(1+z)^2
ZTrans15Prime(z) = 1 + 1/(1+z)^2
ZTrans16Prime(z) = 1 + 1/(1-z)^2
ZTrans17Prime(z) = 1 - 1/(1-z)^2
ZTrans18Prime(z) = 1 + 1/(1+z)^2
ZTrans19Prime(z) = 1 - 1/(1+z)^2
ZTrans20Prime(z) = 1 + 1/(1-z)^2
ZTrans21Prime(z) = 1 - 1/(1-z)^2
ZTrans22Prime(z) = -1/(1+z)^2 + 1/(1-z)^2
ZTrans23Prime(z) = -1/(1+z)^2 - 1/(1-z)^2
ZTrans24Prime(z) = 1/(1+z)^2 + 1/(1-z)^2
ZTrans25Prime(z) = 1/(1+z)^2 - 1/(1-z)^2
ZTrans26Prime(z) = -1/(1+z)^2 + 1/(1-z)^2
ZTrans27Prime(z) = -1/(1+z)^2 - 1/(1-z)^2
ZTrans28Prime(z) = 1/(1+z)^2 + 1/(1-z)^2
ZTrans29Prime(z) = 1/(1+z)^2 - 1/(1-z)^2
ZTrans30Prime(z) = z*(2+z)/(1+z)^2
ZTrans31Prime(z) = z*(2-z)/(1-z)^2
ZTrans32Prime(z) = -(2+z)/z^3
ZTrans33Prime(z) = -(2-z)/z^3
ZTrans34Prime(z) = 2*z - 1/z^2
ZTrans35Prime(z) = 2*z + 1/z^2
ZTrans36Prime(z) = 2*z - 1/(1+z)^2
ZTrans37Prime(z) = 2*z + 1/(1+z)^2
ZTrans38Prime(z) = 2*z + 1/(1-z)^2
ZTrans39Prime(z) = 2*z - 1/(1-z)^2
ZTrans40Prime(z) = 2*z + 1/(1+z)^2
ZTrans41Prime(z) = 2*z - 1/(1+z)^2
ZTrans42Prime(z) = 2*z + 1/(1-z)^2
ZTrans43Prime(z) = 2*z - 1/(1-z)^2
ZTrans44Prime(z) = 1 + 2*z
ZTrans45Prime(z) = 1 - 2*z
ZTrans46Prime(z) = 2*z - 1
ZTrans47Prime(z) = 2*z
ZTrans48Prime(z) = 2*z
ZTrans49Prime(z) = -2*z
ZTrans50Prime(z) = -2*z

2nd Derivatives:

ZTrans1DoublePrime(z) = 2/z^3
ZTrans2DoublePrime(z) = 2/(1+z)^3
ZTrans3DoublePrime(z) = 2/(1-z)^3
ZTrans4DoublePrime(z) = -2/(1+z)^3
ZTrans5DoublePrime(z) = 2/(1-z)^3
ZTrans6DoublePrime(z) = 2/z^3
ZTrans7DoublePrime(z) = 2/z^3
ZTrans8DoublePrime(z) = 4/(z-1)^3
ZTrans9DoublePrime(z) = -4/(z-1)^3
ZTrans10DoublePrime(z) = -4/(z+1)^3
ZTrans11DoublePrime(z) = 4/(z+1)^3
ZTrans12DoublePrime(z) = 2/z^3
ZTrans13DoublePrime(z) = -2/z^3
ZTrans14DoublePrime(z) = 2/(1+z)^3
ZTrans15DoublePrime(z) = -2/(1+z)^3
ZTrans16DoublePrime(z) = 2/(1-z)^3
ZTrans17DoublePrime(z) = -2/(1-z)^3
ZTrans18DoublePrime(z) = -2/(1+z)^3
ZTrans19DoublePrime(z) = 2/(1+z)^3
ZTrans20DoublePrime(z) = 2/(1-z)^3
ZTrans21DoublePrime(z) = -2/(1-z)^3
ZTrans22DoublePrime(z) = 2/(1+z)^3 + 2/(1-z)^3
ZTrans23DoublePrime(z) = 2/(1+z)^3 - 2/(1-z)^3
ZTrans24DoublePrime(z) = -2/(1+z)^3 + 2/(1-z)^3
ZTrans25DoublePrime(z) = -2/(1+z)^3 - 2/(1-z)^3
ZTrans26DoublePrime(z) = 2/(1+z)^3 + 2/(1-z)^3
ZTrans27DoublePrime(z) = 2/(1+z)^3 - 2/(1-z)^3
ZTrans28DoublePrime(z) = -2/(1+z)^3 + 2/(1-z)^3
ZTrans29DoublePrime(z) = -2/(1+z)^3 - 2/(1-z)^3
ZTrans30DoublePrime(z) = 2/(1+z)^3
ZTrans31DoublePrime(z) = 2/(1-z)^3
ZTrans32DoublePrime(z) = (4+z)/z^4
ZTrans33DoublePrime(z) = (4-z)/z^4
ZTrans34DoublePrime(z) = 2 + 2/z^3
ZTrans35DoublePrime(z) = 2 - 2/z^3
ZTrans36DoublePrime(z) = 2 + 2/(1+z)^3
ZTrans37DoublePrime(z) = 2 - 2/(1+z)^3
ZTrans38DoublePrime(z) = 2 + 2/(1-z)^3
ZTrans39DoublePrime(z) = 2 - 2/(1-z)^3
ZTrans40DoublePrime(z) = 2 - 2/(1+z)^3
ZTrans41DoublePrime(z) = 2 + 2/(1+z)^3
ZTrans42DoublePrime(z) = 2 + 2/(1-z)^3
ZTrans43DoublePrime(z) = 2 - 2/(1-z)^3
ZTrans44DoublePrime(z) = 2
ZTrans45DoublePrime(z) = -2
ZTrans46DoublePrime(z) = 2
ZTrans47DoublePrime(z) = 2
ZTrans48DoublePrime(z) = 2
ZTrans49DoublePrime(z) = -2
ZTrans50DoublePrime(z) = -2

Inverse Functions:

InverseZTrans1(z) = 1/z
InverseZTrans2(z) = (1-z)/z
InverseZTrans3(z) = (z-1)/z
InverseZTrans4(z) = z/(1-z)
InverseZTrans5(z) = z/(1+z)
InverseZTrans6(z) = 1/(z-1)
InverseZTrans7(z) = 1/(1+z)
InverseZTrans8(z) = (z+1)/(z-1)
InverseZTrans9(z) = (z-1)/(z+1)
InverseZTrans10(z) = (1+z)/(1-z)
InverseZTrans11(z) = (1-z)/(z+1)

InverseZTrans12(z) = (Root(z^2-4, 2, Random.Integer(2))+z)/2
InverseZTrans13(z) = (Root(z^2+4, 2, Random.Integer(2))+z)/2
InverseZTrans14(z) = (Root(z^2+2*z-3, 2, Random.Integer(2))+1-z)/2
InverseZTrans15(z) = (Root(z^2+2*z+5, 2, Random.Integer(2))+1-z)/2
InverseZTrans16(z) = (Root(z^2-2*z+5, 2, Random.Integer(2))-1-z)/2
InverseZTrans17(z) = (Root(z^2-2*z-3, 2, Random.Integer(2))-1-z)/2
InverseZTrans18(z) = (Root(z^2+4, 2, Random.Integer(2))+2-z)/2
InverseZTrans19(z) = (Root(z^2+4*z, 2, Random.Integer(2))+z)/2
InverseZTrans20(z) = (Root(z^2+4, 2, Random.Integer(2))-2-z)/2
InverseZTrans21(z) = (Root(z^2-4*z, 2, Random.Integer(2))+z)/2
InverseZTrans22(z) = Root(1-2/z, 2, Random.Integer(2))
InverseZTrans23(z) = (Root(z^2+1, 2, Random.Integer(2))+1)/z
InverseZTrans24(z) = (Root(z^2-2*z+2, 2, Random.Integer(2))-1)/(z-1)
InverseZTrans25(z) = Root((z+1)/(z-1), 2, Random.Integer(2))
InverseZTrans26(z) = Root((z-1)/(z+1), 2, Random.Integer(2))
InverseZTrans27(z) = (Root(z^2-2*z+2, 2, Random.Integer(2))+1)/(z-1)
InverseZTrans28(z) = (Root(z^2+1, 2, Random.Integer(2))-1)/z
InverseZTrans29(z) = Root(z/(z-2), 2, Random.Integer(2))
InverseZTrans30(z) = (Root(z^2+4*z, 2, Random.Integer(2))+z)/2
InverseZTrans31(z) = (Root(z^2+4*z, 2, Random.Integer(2))-z)/2
InverseZTrans32(z) = (Root(4*z+1, 2, Random.Integer(2))+1)/(2*z)
InverseZTrans33(z) = (Root(4*z+1, 2, Random.Integer(2))-1)/(2*z)
InverseZTrans44(z) = (Root(4*z+1, 2, Random.Integer(2))-1)/2
InverseZTrans45(z) = (Root(1-4*z, 2, Random.Integer(2))+1)/2
InverseZTrans46(z) = (Root(4*z+1, 2, Random.Integer(2))+1)/2
InverseZTrans47(z) = Root(z-1, 2, Random.Integer(2))
InverseZTrans48(z) = Root(z+1, 2, Random.Integer(2))
InverseZTrans49(z) = Root(1-z, 2, Random.Integer(2))
InverseZTrans50(z) = Root(-z-1, 2, Random.Integer(2))

Note that inverse functions for ZTrans34 through ZTrans43 are excluded since the inverse functions are very complicated.

The functions InverseZTrans12 through InverseZTrans50 return one of two possible results, at random, for a given argument z, so they are normally used with Orbital fractals.

 

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