An exponential like function for rational indices is proposed & introduced in this PPT. While the exponential function is for integer indices and emulates the growth (or decay) type of linear differential equations, the proposed function emulates a class of non-linear differential equations. Exponential function is one of the special case of the proposed function (when m=1).

1. EXPONENTIAL LIKE FUNCTION
FOR RATIONAL INDICES
By Sreeni C, PMP, B.Tech (IITB), MBA29-Aug-2015

2. Function Definition
By Sreeni C (schopakatla@gmail.com)
Definition: A two variable function f(x,m), where x is a positive real number
and m is a natural number
f(x,m) is an infinite series, wherein N+1th Term of the
Series=
𝑥
𝑁+
1
𝑚
1
𝑚 ∗ 1+1
𝑚 ∗ 2+1
𝑚 ∗⋯∗ 𝑁+1
𝑚
𝑁=∞
𝑁=0
ƒ(x,m)=
𝑥
1
𝑚
1
𝑚
+
𝑥
1+
1
𝑚
1
𝑚 ∗ 1+1
𝑚
+
𝑥
2+
1
𝑚
1
𝑚 ∗ 1+1
𝑚 ∗ 2+1
𝑚
+……..+
𝑥
𝑁+
1
𝑚
1
𝑚 ∗ 1+1
𝑚 ∗ 2+1
𝑚 ∗⋯∗ 𝑁+1
𝑚
… . ∞

3. Function Continuity Test
By Sreeni C (schopakatla@gmail.com)
ƒ(x,m) can be expressed as a product of the following two functions
ƒ(x,m)=g*h
Where
g(x,m)=
𝑥
1
𝑚
1
𝑚
h x, m = 1 +
x
1+1
𝑚
+
𝑥2
1+1
𝑚 ∗ 2+1
𝑚
+……..+
𝑥 𝑁
1+1
𝑚 ∗ 2+1
𝑚 ∗⋯∗ 𝑁+1
𝑚
… . ∞
For all positive real numbers (x) and natural numbers (m), g is a
continuous function; h is a polynomial function and hence is a continuous
function for all real numbers
Therefore f=g*h is a continuous function for all positive real numbers
(x) and natural numbers (m)

4. Function Convergence Test
By Sreeni C (schopakatla@gmail.com)
N+2th term of the series (TN+2)=
𝑥
1
𝑚
1
𝑚
*
𝑥 𝑁+1
1+1
𝑚 ∗ 2+1
𝑚 ∗⋯∗ 𝑁+1+1
𝑚
N+1th term of the series(TN+1)=
𝑥
1
𝑚
1
𝑚
*
𝑥 𝑁
1+1
𝑚 ∗ 2+1
𝑚 ∗⋯∗ 𝑁+1
𝑚
TN+2/ TN+1 =
x
(𝑁+1+
1
𝑚
)
lim
𝑁→∞
TN+2/ TN+1 =0 for all finite positive real numbers (x). Hence ƒ(x,m) is a
convergent function for all finite positive real numbers.

5. Function Upper Bound
By Sreeni C (schopakatla@gmail.com)
Upper Bound
ƒ(x,m) can be expressed as a product of the following two functions
ƒ(x,m)=g*h
Where
g(x,m)=
𝑥
1
𝑚
1
𝑚
h x, m = 1 +
x
1+1
𝑚
+
𝑥2
1+1
𝑚 ∗ 2+1
𝑚
+……..+
𝑥 𝑁
1+1
𝑚 ∗ 2+1
𝑚 ∗⋯∗ 𝑁+1
𝑚
… . ∞
h x, m ≤ 1 +
x
1
+
𝑥2
1∗2
+……..+
𝑥 𝑁
1∗2∗⋯∗𝑁
… . ∞≡ h x, m ≤ 𝑒 𝑥
Hence ƒ(x,m) ≤
𝑥
1
𝑚
1
𝑚
x 𝑒 𝑥≡ ƒ(x,m) ≤ mx1/𝑚 𝑒 𝑥

6. Function Lower Bound
By Sreeni C (schopakatla@gmail.com)
Lower Bound
h x, m ≥ 1 +
x
2!
+
𝑥2
3!
+……..+
𝑥 𝑁
𝑁+1 !
… . ∞≡ h x, m ≥ (𝑒 𝑥−1)/𝑥
Hence ƒ(x,m)≥
𝑥
1
𝑚
1
𝑚
x((𝑒 𝑥
−1)/𝑥)≡ ƒ(x,m)≥mx1/𝑚
((𝑒 𝑥
−1)/𝑥)
Therefore the function is bounded by the following Upper &
Lower Bounds
mx1/𝑚
𝑒 𝑥
≥ ƒ(x,m) ≥ mx1/𝑚
((𝑒 𝑥
−1)/𝑥)

7. Solution to a Differential Equation
By Sreeni C (schopakatla@gmail.com)
𝜕ƒ(x, m)/∂x=x(
1
𝑚
−1)
+ ƒ(x, m)
The two variable function ƒ(x, m) is the solution for the non-linear
differential of the type
𝛿𝑦
𝛿𝑥
= x(
1
𝑚
−1)
+y, re-expressed as
XY’ −𝑿𝒀 − 𝑿 𝟏/𝒎=0
Where X is positive real number and m is a natural number

8. Function Behavior (X>1)
By Sreeni C (schopakatla@gmail.com)
The following plot displays the series progression for a sample value of X=5 & m=5
Corollary#1 (To be Proved)
The infinite series peaks at the N=round (X) term and starts sloping down thereafter
0
2
4
6
8
10
12
14
16
18
20
0 5 10 15 20 25 30
Series Nth Term
Series Progression (Series Nth Term Value vs Nth Term)
X=5
m=5