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# Category:Theorem

From specialfunctionswiki

## Pages in category "Theorem"

The following 200 pages are in this category, out of 557 total.

(previous page) (next page)### F

- Faber F1 is nowhere differentiable
- Faber F2 is continuous
- Faber F2 is nowhere differentiable
- Fibonacci zeta at 1 is irrational
- Fibonacci zeta in terms of a sum of binomial coefficients
- Fresnel C in terms of erf
- Fresnel C is odd
- Fresnel S in terms of erf
- Fresnel S is odd
- Functional equation for Riemann xi
- Functional equation for Riemann zeta
- Functional equation for Riemann zeta with cosine

### G

- Gamma function written as infinite product
- Gamma'(z)/Gamma(z)=-gamma-1/z+Sum z/(k(z+k))
- Gamma(1)=1
- Gamma(n+1)=n!
- Gamma(z) as integral of a power of log(1/t) for Re(z) greater than 0
- Gamma(z)Gamma(1-z)=pi/sin(pi z)
- Gamma(z+1)=zGamma(z)
- Gelfond constant is transcendental
- Gelfond-Schneider constant is transcendental
- Generating function for Hermite (physicist) polynomials
- Generating function for Laguerre L
- Generating function for partition function
- Generating relation for Bateman F

### H

- H (-(n+1/2))(z)=(-1)^n J (n+1/2)(z) for integer n geq 0
- H (1/2)(z)=sqrt(2/(pi z))(1-cos(z))
- H (3/2)(z)=sqrt(z/(2pi))(1+2/z^2)-sqrt(2/(pi z))(sin(z)+cos(z)/z)
- H (nu)(x) geq 0 for x gt 0 and nu geq 1/2
- Halving identity for cosh
- Halving identity for sinh
- Halving identity for tangent (1)
- Halving identity for tangent (2)
- Halving identity for tangent (3)
- Hankel H (1) in terms of csc and Bessel J
- Hankel H (2) in terms of csc and Bessel J
- Hermite (physicist) polynomial at negative argument
- Hurwitz zeta absolute convergence
- Hyperfactorial in terms of K-function

### I

- Identity written as a sum of Möbius functions
- Integral (t-b)^(x-1)(a-t)^(y-1)dt=(a-b)^(x+y-1)B(x,y)
- Integral from a to a
- Integral of (1+bt^z)^(-y)t^x dt = (1/z)*b^(-(x+1)/z) B((x+1)/z,y-(x+1)/z)
- Integral of (1+t)^(2x-1)(1-t)^(2y-1)(1+t^2)^(-x-y)dt=2^(x+y-2)B(x,y)
- Integral of (t-b)^(x-1)(a-t)^(y-1)/(c-t)^(x+y) dt = (a-b)^(x+y-1)/((c-a)^x (c-b)^y) B(x,y)
- Integral of (t-b)^(x-1)(a-t)^(y-1)/(t-x)^(x+y) dt=(a-b)^(x+y-1)/((a-c)^x(b-c)^y) B(x,y)
- Integral of (z^n)log(z)dz=(z^(n+1)/(n+1))log(z)-z^(n+1)/(n+1)^2 for integer n neq -1
- Integral of Bessel J for nu=1
- Integral of Bessel J for nu=2n
- Integral of Bessel J for nu=2n+1
- Integral of Bessel J for nu=n+1
- Integral of Bessel J for Re(nu) greater than -1
- Integral of inverse erf from 0 to 1
- Integral of log of inverse erf from 0 to 1
- Integral of monomial times Bessel J
- Integral of t^(x-1)(1-t^z)^(y-1) dt=(1/z)B(x/z,y)
- Integral representation of Airy Ai
- Integral representation of polygamma 2
- Integral representation of polygamma for Re(z) greater than 0
- Integral representation of Struve function
- Integral representation of Struve function (2)
- Integral representation of Struve function (3)
- Integral t^(x-1)(1+bt)^(-x-y) dt = b^(-x) B(x,y)
- Integral t^(x-1)(1-t)^(y-1)(1+bt)^(-x-y)dt = (1+b)^(-x)B(x,y)

### L

- L n'(0)=-n
- L n'(x)=-Sum L k(x)
- L n(0)=1
- L n(x)=(e^x/n!)d^n/dx^n(x^n e^(-x))
- L(-n)=(-1)^nL(n)
- L(n)=F(n+1)+F(n-1)
- L(n)^2-5F(n)^2=4(-1)^n
- L(n+1)L(n-1)-L(n)^2=5(-1)^(n+1)
- Laurent series for log((z+1)/(z-1)) for absolute value of z greater than 1
- Laurent series of the Riemann zeta function
- Legendre chi in terms of Lerch transcendent
- Legendre chi in terms of polylogarithm
- Lerch transcendent polylogarithm
- Li 2(1)=pi^2/6
- Li 2(z)+Li 2(1-z)=pi^2/6-log(z)log(1-z)
- Li 2(z)=-Li 2(1/z)-(1/2)(log z)^2 + i pi log(z) + pi^2/3
- Li2(z)=zPhi(z,2,1)
- Limit of (1/Gamma(c))*2F1(a,b;c;z) as c approaches -m
- Limit of erf when z approaches infinity and the modulus of arg(z) is less than pi/4
- Limit of log(x)/x^a=0
- Limit of q-exponential E sub 1/q for 0 less than q less than 1
- Limit of quotient of consecutive Fibonacci numbers
- Limit of x^a log(x)=0
- Limiting value of Fresnel C
- Limiting value of Fresnel S
- Log 10(z)=log 10(e)log(z)
- Log 10(z)=log(z)/log(10)
- Log a(b)=1/log b(a)
- Log base a in terms of logarithm base b
- Log e(z)=log(z)
- Log((1+z)/(1-z)) as continued fraction
- Log(1+x) less than x
- Log(1+z) as continued fraction
- Log(x) less than or equal to n(x^(1/n)-1)
- Log(x) less than or equal to x-1
- Log(z)=log(10)log 10(z)
- Logarithm (multivalued) of a quotient is a difference of logarithms (multivalued)
- Logarithm (multivalued) of product is a sum of logarithms (multivalued)
- Logarithm (multivalued) of the exponential
- Logarithm at -i
- Logarithm at i
- Logarithm at minus 1
- Logarithm diverges to negative infinity at 0 from right
- Logarithm of 1
- Logarithm of a complex number
- Logarithm of a quotient is a difference of logarithms
- Logarithm of a quotient of Jacobi theta 4 equals a sum of sines
- Logarithm of exponential
- Logarithm of product is a sum of logarithms
- Logarithm of quotient of Jacobi theta 1 equals the log of a quotient of sines + a sum of sines
- Logarithm of quotient of Jacobi theta 2 equals the log of a quotient of cosines + a sum of sines
- Logarithm of quotient of Jacobi theta 3 equals a sum of sines
- Logarithmic derivative of Jacobi theta 1 equals cotangent + a sum of sines
- Logarithmic derivative of Jacobi theta 2 equals negative tangent + a sum of sines
- Logarithmic derivative of Jacobi theta 3 equals a sum of sines
- Logarithmic derivative of Jacobi theta 4 equals a sum of sines
- Logarithmic derivative of Riemann zeta in terms of Mangoldt function
- Logarithmic derivative of Riemann zeta in terms of series over primes

### M

### N

### O

### P

- Partial derivative of beta function
- Pell constant is irrational
- Period of cosh
- Period of sinh
- Period of tanh
- Pi is irrational
- Pochhammer symbol with non-negative integer subscript
- Polygamma multiplication formula
- Polygamma recurrence relation
- Polygamma reflection formula
- Polygamma series representation
- Prime number theorem, logarithmic integral
- Prime number theorem, pi and x/log(x)
- Product of Weierstrass elementary factors is entire
- Product representation of q-exponential E sub 1/q
- Product representation of totient
- Pure recurrence relation for partition function
- Pythagorean identity for coth and csch
- Pythagorean identity for sin and cos
- Pythagorean identity for sinh and cosh
- Pythagorean identity for tanh and sech

### Q

- Q-derivative of q-Cosine
- Q-derivative of q-Sine
- Q-derivative power rule
- Q-difference equation for q-exponential E sub 1/q
- Q-difference equation for q-exponential E sub q
- Q-Euler formula for e sub q
- Q-Euler formula for E sub q
- Q-Gamma at 1
- Q-Gamma at z+1
- Q-number of a negative
- Q-number when a=n is a natural number

### R

- Ramanujan tau inequality
- Ramanujan tau is multiplicative
- Ramanujan tau of a power of a prime
- Real and imaginary parts of log
- Reciprocal gamma is entire
- Reciprocal gamma written as an infinite product
- Reciprocal of i
- Reciprocal of Riemann zeta as a sum of Möbius function for Re(z) greater than 1
- Reciprocal Riemann zeta in terms of Mobius
- Recurrence relation for partition function with sum of divisors
- Recurrence relation for Struve fuction
- Recurrence relation for Struve function (2)
- Recurrence relation of exponential integral E
- Relation between polygamma and Hurwitz zeta
- Relationship between Airy Ai and modified Bessel K
- Relationship between Airy Bi and modified Bessel I
- Relationship between Anger function and Bessel J
- Relationship between arcsin and arccsc
- Relationship between arctan and arccot
- Relationship between Bessel I and Bessel J
- Relationship between Bessel I sub -1/2 and cosh
- Relationship between Bessel I sub 1/2 and sinh
- Relationship between Bessel J and hypergeometric 0F1
- Relationship between Bessel J sub n and Bessel J sub -n
- Relationship between Bessel Y sub n and Bessel Y sub -n
- Relationship between Bessel-Clifford and hypergeometric 0F1
- Relationship between Chebyshev T and Gegenbauer C