Reduction Formulae, Beta and Gamma Functions
Reduction formulae for sinusoidal functions
1. To find reduction formula for $∫ sin^n x dx$, n is positive integer ≥ 2 and evaluate completely $∫↙{0}↖{π/2} sin^n x dx$
$I_n = {n-1}/{n} I_{n-2}$
$\table ∫↙{0}↖{π/2} sin^n x dx,=,{n-1}/n {n-3}/{n-2} …. 3/4 1/2 π/2\; \text"if n is even";,=,{n-1}/n {n-3}/{n-2}…..4/5 2/3 1 \; \text"if n is odd"$
$\table ∫↙{0}↖{π/2} cos^n x dx,=,{n-1}/n {n-3}/{n-2} …. 3/4 1/2 π/2\; \text"if n is even";,=,{n-1}/n {n-3}/{n-2}…..4/5 2/3 1 \; \text"if n is odd"$
Additional Results:
I. $∫↙{0}↖{π} sin^n x dx$ = $2 ∫↙{0}↖{π} sin^n x dx$ , for all n integral values of n
II. $\table ∫↙{0}↖{π} cos^n x dx,=,2 ∫↙{0}↖{π} cos^n x dx \; \text"if n is an even integer";,=,0 \; \text"if n is an odd integer"$
III. $\table ∫↙{0}↖{2π} sin^n x dx,=,4 ∫↙{0}↖{π/2} sin^n x dx \; \text"if n is an even integer";,=,0 \; \text"if n is an odd integer"$
IV. $\table ∫↙{0}↖{2π} cos^n x dx,=,4 ∫↙{0}↖{π/2} cos^n x dx \; \text"if n is an even integer";,=,0 \; \text"if n is an odd integer"$
2. To find a reduction formula for $∫sin^m x cos^nx dx$, where m and n are positive integers ≥ 2 and to completely evaluate $∫↙{0}↖{π/2} sin^m x cos^n x dx$
Note: $∫ [f(x)]^m f'(x) dx = {[f(x)]^{m+1}}/{m+1}$
$I_{m,n} = {n-1}/{m+n} I_{m,n-2}$
$\table ∫↙{0}↖{π/2} sin^m x cos^nx dx,=, {[(m-1)(m-3)….2 \text"or" 1] [(n-1) (n-3)…2 \text"or" 1]}/{(m+n)(m+n-2)(m+n-4)… 2 \text"or" 1} \text"x P"; \text"where P",=,π/2 \text"if m and n are both even";,=,1 \text"for all other values of m and n" $
Additional Results
I. $∫↙{0}↖{π/2} sin^Px cosx dx = 1/{P+1} = ∫↙{0}↖{π/2} cos^Px sinx dx$
II. $\table ∫↙{0}↖{π} sin^mx cos^nx dx,=,2 ∫↙{0}↖{π/2} sin^mx cos^nx dx \text"if n=even, m= even or odd";,=,0 \text"if n = odd, m = even or odd"$
III. $\table ∫↙{0}↖{2π} sin^mx cos^nx dx,=,4 ∫↙{0}↖{π/2} sin^mx cos^nx \text"if m,n = even";,=,0 \text"otherwise"$
3. Reduction formula for $∫ tan^nx dx$
$∫tan^nx dx = {tan^{n-1}x}/{n-1} – ∫ tan^{n-2}x dx$
4. Reduction formula for $∫sec^nθ dθ$
$∫ sec^nθ dθ = {sec^{n-2}θ tanθ}/{n-1} + {n-2}/{n-1} ∫sec^{n-2}θ dθ$
Gamma functions
$|\ov{n} = ∫↙{0}↖{∞} e^{-x} x^{n-1} dx$ (n>0)
a.k.a. Euler’s integral of the second kind
Properties of gamma functions
1. $|\ov{n} = 2 ∫↙{0}↖{∞} e^{-x^2} x^{2n-1} dx$
2. $|\ov{1} =1$
3. Reduction formula for gamma functions
$\table |\ov{(n+1)},=,n | \ov{n} \text", in general";,=,n! \text"if n is a positive integer"$
4. $|\ov{0} = ∞$
5. $|\ov{1/2} = √π$
6. $|\ov{(n+1)} = n!$
7. For negative fraction n, we use
$|\ov{n} = {|\ov{(n+1)}}/{n} |\ov{- 5/3}$
Transformation of gamma functions
1. $ ∫↙{0}↖{∞} e^{-ky} y^{n-1} dy = {|\ov{n}}/{k^n}$
2. $ ∫↙{0}↖{∞} e^{-y^{1/n}} dy = n |\ov{n} = |\ov{(n+1)}$
3. $|\ov{n} = ∫↙{0}↖{1} (log1/y)^{n-1} dy $
Additional Results: $|\ov{P} |\ov{1-P} = {π}/{sin pπ} \text"if 0
Beta function
$B(m,n) = ∫↙{0}↖{1} x^{m-1} (1-x)^{n-1} dx, \text"m>0 n>0"$
Properties of beta functions
1. B(m, n) = B(n, m)
2. $ ∫↙{0}↖{1} x^m (1-x)^n dx = B(m+1, n+1)$
3. B(m, n) = 2 $ ∫↙{0}↖{π/2} sin^{2m-1}θ cos^{2n-1}θ dθ$
Standard formula: $∫↙{0}↖{π/2} sin^pθ cos^qθ dθ = 1/2 B({p+1}/2 , {q+1}/2) $
4. B(m, n) = $∫↙{0}↖{∞} {x^{m-1}}/{(1+x)^{m+n}} dx$
5. Relation between beta and gamma functions
B(m, n) = ${|\ov{m} |\ov{n}}/{|\ov{m+n}}$
Duplication formula of gamma functions
$|\ov{m} |\ov{m + 1/2} = {√π}/{2^{2m – 1}} |\ov{2m}$
Addtional Results: $π/{sinpπ} = |\ov{p} |\ov{1-p}$
Given $I = ∫↙{0}↖{∞} {x^{p-1}}/{1+x} dx = π/{sinpπ}$ for 0 < p < 1