Are there any parent functions other than
f (x) =
0
e^x
Sin (x)
Cos (x)
That through deriving enough, you get back to the original function?
Yes.
A linear combination of the functions you mentioned.
Also cosh and sinh (derivate twice to get back to the same).
Also, think about polynomials. You can set up a function for any n, such that when you differentiate a + bx^n + cx^(2n) + ..., n times, you get back the same function.
Whether or not the gained function is a linear combination of e^x, sinx, cosx, sinhx or coshx, is left as an exercise to the reader..
>>7928915
Interesting! Looking at the hyperbolic trig functions, they seem to be linear combinations of e^x.
>>7928937
as are cos x and sin x
https://en.wikipedia.org/wiki/Parent_function
mhm... , I've a PhD but never even heard of that concept. Is it something American like PEMDAS?
Consider the diffenntial equation:
[eqn] f^{(n)}(t) = f(t)[/eqn]
Applying the Laplace transform gives
[eqn] s^n F(s) - \sum_{k=1}^{n} s^{n-k} f^{(k-1)}(0) = F(s) [/eqn]
[eqn] F(s) = \frac{\sum_{k=1}^{n} s^{n-k} f^{(k-1)}(0)}{s^n - 1} [/eqn]
Choose any values for [math]f^{(k-1)}(0) [/math] and transform back to get a functions whose n-th derivative is itself.
>>7929079
Since the transform is linear this simplifies to linear combinations of the inverse Laplace transforms of s^m/(s^n-1) with 0<= m < n.
For a specific n this can be written in terms of e^at, the gamma function ,and the incomplete gamma function.
For n = 1,2,3 it simplifies to only linear combinations of the exponent function.
>>7928905
Who keeps teaching that "deriving" is the word to use for differentiation? You all realize that "derive" already has a different meaning?
>>7928962
Yeah, they make us learn like 7 parent functions in math class, which are ones like linear, exponential, quadratic, absolute value, ect. It's not really rigorous or anything
Hyperbolic trig functions.
