a^(m+n)

Input

a^(m + n)

Root

a = 0, Re(n)>-Re(m)

Periodicity

periodic in m with period (2 i π)/log(a) periodic in n with period (2 i π)/log(a)

Series expansion at m=0

a^n + m a^n log(a) + 1/2 m^2 a^n log^2(a) + 1/6 m^3 a^n log^3(a) + 1/24 m^4 a^n log^4(a) + O(m^5) (Taylor series)
Big‐O notation »

Derivative

d/dm(a^(m + n)) = log(a) a^(m + n)

Indefinite integral

integral a^(m + n) dm = a^(m + n)/log(a) + constant

Series representation

a^(m + n) = sum_(ν=0)^∞ binomial(m + n, ν) (-1 + a)^ν
a^(m + n) = sum_(ν=0)^∞ (m^ν a^n log^ν(a))/(ν!)
a^(m + n) = sum_(ν=0)^∞ (n^ν a^m log^ν(a))/(ν!)
POWERED BY THE WOLFRAM LANGUAGE