جمعه ۲۷ آبان ۱۳۷۹ ه‍.ش.

کل فرمول های ریاضیات

کل فرمول های ریاضیات انتخاب کنید


Some of these functions I have seen defined under both intervals (0 to x) and (x to inf). In that case, both variant definitions are listed.
gamma = Euler's constant = 0.5772156649...
(x) = Gamma(x) = (integral)(0 to inf)t^(x-1) e^(-t)dt (Gamma function)
B(x,y) = (integral)(0 to 1)t^(x-1) (1-t)^(y-1)DT
(Beta function)
Ei(x) = (integral)(x to inf)e^(-t)/t DT (exponential integral) or it's variant, NONEQUIVALENT form:

Ei(x) = + ln(x) + (integral)(0 to x)(e^t - 1)/t DT = gamma + ln(x) + (sum)(n=1..inf)x^n/(n*n!)
li(x) = (integral)(2 to x)1/ln(t) DT (logarithmic integral)
Si(x) = (integral)(x to inf)sin(t)/t DT (sine integral) or it's variant, NONEQUIVALENT form:

Si(x) = (integral)(0 to x)sin(t)/t DT = PI/2 - (integral)(x to inf)sin(t)/t DT

Ci(x) = (integral)(x to inf)cos(t)/t DT (cosine integral) or it's variant, NONEQUIVALENT form:

CI(x) = - (integral)(x to inf)COs(t)/t DT = gamma + ln(x) + (integral)(0 to x) (COs(t) - 1) / t DT (cosine integral)

Chi(x) = gamma + ln(x) + (integral)(0 to x)(cosh(t)-1)/t DT (hyperbolic cosine integral)
Shi(x) = (integral)(0 to x)sinh(t)/t DT (hyperbolic sine integral)
Erf(x) = 2/PI^(1/2)(integral)(0 to x)e^(-t^2) DT = 2/sqrtPI (sum)(n=0..inf) (-1)^n x^(2n+1) / ( n! (2n+1) ) (error function)
FresnelC(x) = (integral)(0 to x)COs(PI/2 t^2) DT
FresnelS(x) = (integral)(0 to x)sin(PI/2 t^2) DT
dilog(x) = (integral)(1 to x)ln(t)/(1-t) DT
Psi(x) = (d/dx)ln(Gamma(x))
Psi(n,x) = nth derivative of Psi(x)
W(x) = inverse of x*e^x
L sub n (x) = (e^x/n!)( x^n e^(-x) ) (n) (laguerre polynomial degree n. (n) meaning nth derivative)
Zeta(s) = (sum)(n=1..inf) 1/n^s

Dirichlet's beta function B(x) = (sum)(n=0..inf) (-1)^n / (2n+1)^x
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Integral Identities
(Math | Calculus | Integrals | Identities)
Formal Integral Definition:
(integral)(a to b) f(x) dx = lim (d -> 0) (sum) (k=1..n) f(X(k)) (x(k) - x(k-1)) when...
a = x0 <>1 <>2 < ... <>n = b
d = max (x1-x0, x2-x1, ... , xn - x(n-1))
x(k-1) <= X(k) <= x(k) k = 1, 2, ... , n
(integral)(a to b) F '(x) dx = F(b) - F(a) (Fundamental Theorem for integrals of derivatives)

(integral)a f(x) dx = a(integral) f(x) dx (if a is constant)
(integral)f(x) + g(x) dx = (integral)f(x) dx + (integral)g(x) dx
(integral)(a to b) f(x) dx = (integral)f(x) dx | (a b)
(integral)(a to b) f(x) dx + (integral)(b to c) f(x) dx = (integral)(a to c) f(x) dx
(integral)f(u) du/dx dx = (integral)f(u) du (integration by substitution)
فرمول های انتگرال کاملا صحیح برای علاقه مندان ریاضی

Power of x.
(integral)xn dx = x(n+1) / (n+1) + C
(n -1) Proof
(integral)1/x dx = ln|x| + C
Exponential / Logarithmic
(integral)ex dx = ex + C
Proof
(integral)bx dx = bx / ln(b) + C
Proof, Tip!
(integral)ln(x) dx = x ln(x) - x + C
Proof

Trigonometric
(integral)sin x dx = -cos x + C
Proof
(integral)csc x dx = - ln|CSC x + cot x| + C
Proof
(integral)COs x dx = sin x + C
Proof
(integral)sec x dx = ln|sec x + tan x| + C
Proof
(integral)tan x dx = -ln|COs x| + C
Proof
(integral)cot x dx = ln|sin x| + C
Proof
Trigonometric Result
(integral)COs x dx = sin x + C
Proof
(integral)CSC x cot x dx = - CSC x + C
Proof
(integral)sin x dx = COs x + C
Proof
(integral)sec x tan x dx = sec x + C
Proof
(integral)sec2 x dx = tan x + C
Proof
(integral)csc2 x dx = - cot x + C
Proof
Inverse Trigonometric
(integral)arcsin x dx = x arcsin x + sqrt(1-x2) + C
(integral)arccsc x dx = x arccos x - sqrt(1-x2) + C
(integral)arctan x dx = x arctan x - (1/2) ln(1+x2) + C
Inverse Trigonometric Result

(integral)
dx

sqrt(1 - x2)
= arcsin x + C

(integral)
dx

x sqrt(x2 - 1)
= arcsec|x| + C

(integral)
dx

1 + x2
= arctan x + C


Useful Identities
arccos x = pi/2 - arcsin x
(-1 <= x <= 1)
arccsc x = pi/2 - arcsec x
(|x| >= 1)
arccot x = pi/2 - arctan x
(for all x)
Hyperbolic
(integral)sinh x dx = cosh x + C
Proof
(integral)csch x dx = ln |tanh(x/2)| + C
Proof
(integral)cosh x dx = sinh x + C
Proof
(integral)sech x dx = arctan (sinh x) + C
(integral)tanh x dx = ln (cosh x) + C
Proof
(integral)coth x dx = ln |sinh x| + C
Proof

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