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║ Mathematik Formules ║
║ ║
║ By Volpone of Malorean Effect for crazy mathematicians like Toto ║
║ ║
║ Bibliographie : -Fractint.doc (of Fractint 17.1) ║
║ -Pour l'honneur de l'esprit humain ║
║ (les mathématiques aujourd'hui) ║
║ Jean Dieudonné ║
║ ║
║ Version 2.0 ║
║ Release January 1994 ║
║ ║
╚════════════════════════════════════════════════════════════════════════╝
--------------------------- Constantes Part ---------------------------------
π =3.14159265359
π/2 =1.5707963268
2*π =6.28318530718
e =2.71828182846
----------------------- Complexe definitions Part -------------------------------
Z=x+i*y
─
Z=x-i*y
Z=r*e^(i*α) r=√(x^2+y^2) Tan(α)=y/x
Re(Z) = x Im(Z) = y
----------------------- Complexe formules Part -------------------------------
─
Z^-1=Z/(│Z│^2)
│Z+Z'│<=│Z│+│Z'│
─
Z*Z=│Z│^2
n n
√(Z)= √(r)*e^i*(α/n)
│n │ n
│ √(Z)│ = √│Z│ = (x^2+y^2)^(1/(2*n))
Z^n=1 -> Z=e^i*(α/n)
----------------------- Z1=√(Z2) Part -------------------------------
Z1=√(Z2)
Re(Z1)=[√(│Z2│+X2)]/4 ║ Im(Z1)=[√(│Z2│-X2)]/4
Re(Z1)/Im(Z1)=(│Z2│+x2)/│y2│
Re(Z1)*Im(Z1)=│y2│/16
------------------ Ln \ Log \ Exp \ Z^C Part ------------------------
*
xεR ln(x+iy) = (1/2)ln(x*x + y*y) + i(arctan(y/x) + 2kPi)
(k = 0, +-1, +-2, +-....)
Ln(i*y) (y>0) = Ln(y)+i*π/2
(y<0) = Ln(y)-i*π/2
z^z = e^(log(z)*z)
e^i*a = Cos(a) + i*Sin(a)
e^(x+iy) = e^x * e^(i*y)
= (Ch(x) + Sh(x)) * (cos(y) + i*sin(y))
= e^x * (cos(y) + i*sin(y))
= (e^x * cos(y)) + i(e^x * sin(y))
Z=x+i*y C=a+i*b
*
xεR │Z│^a
Z^C = ----------------- * e^i*(a*ArcTan(y/x) + b*Ln│Z│ )
e^(ArcTan(y/x))
-------------------- Circular real function Part --------------------------
xεR yεR
Sh(x) = [e^x - e^(-x)]/2
Ch(x) = [e^x + e^(-x)]/2
Th(x) =
Sh(x)+Ch(x) = e^x
Sin (x+y) = Sin(x)*Cos(y)+Cos(x)*Sin(y)
Sin (x-y) = Sin(x)*Cos(y)-Cos(x)*Sin(y)
Cos (x+y) = Cos(x)*Cos(y)-Sin(x)*Sin(y)
Cos (x-y) = Cos(x)*Cos(y)+Sin(x)*Sin(y)
Tan (x+y) = [Tan(x)+Tan(b)] / [1-Tan(x)*Tan(y)]
Tan (x-y) = [Tan(x)-Tan(b)] / [1+Tan(x)*Tan(y)]
ATan(x+y) =
------------------- Circular complex function Part ------------------------
sin (x+iy) = sin(x)Ch(y) + icos(x)Sh(y)
cos (x+iy) = cos(x)Ch(y) - isin(x)Sh(y)
Sh(x+iy) = Sh(x)cos(y) + iCh(x)sin(y)
Ch(x+iy) = Ch(x)cos(y) + iSh(x)sin(y)
sin(2x) Sh(2y)
tan(x+iy) = ------------------ + i------------------
cos(2x) + Ch(2y) cos(2x) + Ch(2y)
Sh(2x) sin(2y)
tanh(x+iy) = ------------------ + i------------------
Ch(2x) + cos(2y) Ch(2x) + cos(2y)
sin(2x) - i*Sh(2y)
cotan(x+iy) = --------------------
Ch(2y) - cos(2x)
Sh(2x) - i*sin(2y)
cotanh(x+iy) = --------------------
Ch(2x) - cos(2y)
-------------------- Expoly1 Part -----------------------------
│ The Expoly1 formule is use for the matematica 2 │
│ Expoly1 is create by Volpone Of Malorean Effect │
------------------------------------------------------------------
P(x)=A.x^2+B.X+C
d
P'(x) = --- P(x) = 2*A.x+B
dx
d
P''(x) = --- P'(x) = 2*A
dx
d
P'''(x) = --- P''(x) = 0
dx
-The formule
d
Fx(0)=P(X) Fx(N) = Fx(N-1) + ----Fx(N-1)
dx
-Example :first number of suite
(Fx(1)=P(x)+P'(x) )
d
(Fx(2)=P(x)+P'(x) + ---[P(x)+P'(x)]
dx
=P(x)+P'(x)+[P'(x)+P''(x)]
=P(x)+2*P'(x)+P''(x)
- The formules
N-1
Fx(N)=A.x^2 + (B+2*A*N).X + C+B*N+2*A*Σ i
i=1
N-1
Fx(N)=P(x) + N*P'(X) + Σ i *P''(x)
i=1
Fx(N+1)=Fx(N)+P'(x)+N*P''(x)
Exemples : F(x,y) = Px(x) + Px(y) = Color to pixel(x,y)
---------------------------- Vector Part -------------------------------
-> │Ux -> │Vx
U │Uy V │Vy
│Uz │Vy
-> -> ║ -> -> │Uy*Vz-Vy*Uz
U ∙ V = Ux*Vx+Uy*Vy+Uz*Vz ║ U ^ V = │Uz*Vx-Vz*Ux
║ │Ux*Vy-Vx*Uy
-> -> -> -> -> -> -> -> ->
U ^ (V ^ W ) = (U . W )*V - (U . V )*W
----------------------- Quaternion formules Part -------------------------------
Quarternions has discover by Hamilton in 1843
On pose Qu = ensembles nombres quarternions
Q εQu -> Q1=A +i*B +j*C +k*D ╔══════════╦═════════╦════════╦════════╗
Q1εQu -> Q1=A1+i*B1+j*C1+k*D1 ║ i^2=-1 ║ i*j=+k ║ i*k=-j ║ j*k=+i ║
Q2εQu -> Q2=A2+i*B2+j*C2+k*D2 ║ k^2=-1 ║ j*i=-k ║ k*i=+j ║ k*j=-i ║
╚══════════╩═════════╩════════╩════════╝
│Q│=√(A^2+B^2+C^2+D^2)
Q1+Q2=(A1+A2)+(B1+B2)*i+(C1+C2)*j+(D1+D2)*k
Q1*Q2= (A1*A2-B1*B2-C1*C2-D1*D2)
+(A1*B2+B1*A2+C1*D2+D1*C2)*i
+(A1*C2-B1*D2+C1*A2+D1*B2)*j
+(A1*D2+B1*C2-C1*B2+D1*D3)*k
ZORG !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
