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) = t^{^(x-1)} e^{^(-t)}dt (Gamma function)
B(x,y) = t^{^(x-1)} (1-t)^{^(y-1)}DT (Beta function)
Ei(x) = e^{^(-t)}/t DT (exponential integral) or it's variant, NONEQUIVALENT form:

Ei(x) = + ln(x) + (e^{^t} - 1)/t DT = gamma + ln(x) + (n=1..inf)x^{^n}/(n*n!)

li(x) = 1/ln(t) DT (logarithmic integral)
Si(x) = sin(t)/t DT (sine integral) or it's variant, NONEQUIVALENT form:

Si(x) = sin(t)/t DT = PI/2 - sin(t)/t DT

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

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

Chi(x) = gamma + ln(x) + (cosh(t)-1)/t DT (hyperbolic cosine integral)
Shi(x) = sinh(t)/t DT (hyperbolic sine integral)
Erf(x) = 2/PI^{^(1/2)}e^{^(-t^2)} DT = 2/PI (n=0..inf) (-1)^{^n} x^{^(2n+1)} / ( n! (2n+1) ) (error function)
FresnelC(x) = COs(PI/2 t^{^2}) DT
FresnelS(x) = sin(PI/2 t^{^2}) DT
dilog(x) = ln(t)/(1-t) DT
Psi(x) = ln(Gamma(x))
Psi(n,x) = n^th 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 n^th derivative)
Zeta(s) = (n=1..inf) 1/n^{^s}
Dirichlet's beta function B(x) = (n=0..inf) (-1)^{^n} / (2n+1)^{^x}
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Integral Identities

(Math | Calculus | Integrals | Identities)

Formal Integral Definition:
f(x) dx = lim _{(d -> 0)} (k=1..n) f(X_(k)) (x_(k) - x_(k-1)) when...

a = x₀ <>1 <>2 < ... <>n = b

d = max (x₁-x₀, x₂-x₁, ... , x_n - x_(n-1))

x_(k-1) <= X_(k) <= x_(k) k = 1, 2, ... , n

F '(x) dx = F(b) - F(a) (Fundamental Theorem for integrals of derivatives)

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

Power of x.

xn dx = x(n+1) / (n+1) + C (n -1) Proof

1/x dx = ln|x| + C

Exponential / Logarithmic

ex dx = ex + C Proof	bx dx = bx / ln(b) + C Proof, Tip!
ln(x) dx = x ln(x) - x + C Proof

Trigonometric

sin x dx = -cos x + C Proof	csc x dx = - ln|CSC x + cot x| + C Proof
COs x dx = sin x + C Proof	sec x dx = ln|sec x + tan x| + C Proof
tan x dx = -ln|COs x| + C Proof	cot x dx = ln|sin x| + C Proof

Trigonometric Result

COs x dx = sin x + C Proof	CSC x cot x dx = - CSC x + C Proof
sin x dx = COs x + C Proof	sec x tan x dx = sec x + C Proof
sec2 x dx = tan x + C Proof	csc2 x dx = - cot x + C Proof

Inverse Trigonometric

arcsin x dx = x arcsin x + (1-x2) + C

arccsc x dx = x arccos x - (1-x2) + C

arctan x dx = x arctan x - (1/2) ln(1+x2) + C

Inverse Trigonometric Result

dx (1 - x2) = arcsin x + C dx x (x2 - 1) = arcsec|x| + C dx 1 + x2 = arctan x + C

Useful Identities arccos x = /2 - arcsin x (-1 <= x <= 1) arccsc x = /2 - arcsec x (|x| >= 1) arccot x = /2 - arctan x (for all x)

Hyperbolic

sinh x dx = cosh x + C Proof	csch x dx = ln |tanh(x/2)| + C Proof
cosh x dx = sinh x + C Proof	sech x dx = arctan (sinh x) + C
tanh x dx = ln (cosh x) + C Proof	coth x dx = ln |sinh x| + C Proof

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