# A Model of Information Visualization Interpretation
## 3 Results
### The Mean and Useful Properties  
By definition,  
\[\text{E}(D_{12}/\psi_{A})=\int_{0}^{\infty}yg(y)dy. \tag{26}\][PERSON] determined this mean [29]. Multiplying by \(\psi_{A}\), he found  
\[\mu=\mathrm{E}(D_{12})=\psi_{A}\sqrt{2}\exp\left(-\frac{h^{2}}{2\psi_{A}^{2}} \right)\frac{\Gamma(3/2)}{\Gamma(1)}\,_{1}\mathrm{F}_{1}\left(\frac{3}{2};1; \frac{h^{2}}{2\psi_{A}^{2}}\right) \tag{27}\]  
where \({}_{1}\mathrm{F}_{1}(a;b;x)\) is the confluent hypergeometric function. It is also denoted as \(\mathrm{M}(a,b,x)\) by several important authors. It is labeled this way to be consistent with generalized hypergeometric functions.  
There will be need for the power series expansion of \({}_{1}\mathrm{F}_{1}(a;b;x)\). For \(z\in\mathbb{C}\),  
\[{}_{1}\mathrm{F}_{1}(a;b;z)=\sum_{s=0}^{\infty}\frac{(a)_{s}}{(b)_{s}s!}z^{s}= 1+\frac{a}{b}z+\frac{a(a+1)}{b(b+1)2!}z^{2}+\ldots\quad(b\
eq 0,-1,-2,-3,\ldots) \tag{28}\]  
where \((a)_{s}\) is the Pochhammer symbol with the properties  
\[(a)_{0} =1\] \[(a)_{n} =a(a+1)(a+2)\ldots(a+n-1)=\frac{\Gamma(a+n)}{\Gamma(a)}.\]  
The Gamma function is used repeatedly, so a few useful properties are given for the convenience of the reader:  
\[\Gamma(n) =(n-1)! n=1,2,3,\ldots\] \[\Gamma(z+1) =z\Gamma(z) z\in\mathbb{C}\] \[\Gamma(1/2) =\sqrt{\pi}\] \[\Gamma(3/2) =\sqrt{\pi}/2.\]  
As the confluent hypergeometric function is less common and yet critical to the analysis, several of its basic properties are given. Then, a simple lemma encapsulating useful attributes is proven. Many features of this function are available in the _NIST Handbook of Mathematical Functions_[30]. (The numbers below, such as (13.2.39), locate the given equation in the sizable NIST reference [30] and are provided for ease of reference for those who wish to explore other properties of this remarkable function).  
1. The power series in Equation (28) for \({}_{1}\mathrm{F}_{1}(a;b;z)\) is entire in \(z\) and \(a\) and meromorphic in \(b\) for \(b\
eq 0,-1,-2,\ldots\);
2. [PERSON]'s theorem states \[{}_{1}\mathrm{F}_{1}(a;b;z)=e^{z}\,{}_{1}\mathrm{F}_{1}(b-a;b;-z)\quad(13.2. 39);\] (29)
3. Recurrence relationship \[b\,{}_{1}\mathrm{F}_{1}(a;b;z)-b\,{}_{1}\mathrm{F}_{1}(a-1;b;z)-z\,{}_{1} \mathrm{F}_{1}(a;b+1;z)=0\quad(13.3.4);\] (30)
4. Derivative formulae \[\frac{d}{dz}\,{}_{1}\mathrm{F}_{1}(a;b;z)=\frac{a}{b}\,{}_{1}\mathrm{F}_{1}(a+ 1;b+1;z)\quad(13.3.15)\] (31) and \[\frac{d}{dz}(e^{-z}\,{}_{1}\mathrm{F}_{1}(a;b;z))=(-1)^{1}\frac{(b-a)_{1}}{(b )_{1}}e^{-z}\,{}_{1}\mathrm{F}_{1}(a;b+1;z)\quad(13.3.20).\] (32) Now, we provide a lemma concerning \({}_{1}\mathrm{F}_{1}(a;b;x)\) that is used extensively.  
**Lemma 1**.: _For \(a,b\in\mathbb{R}\) such that \(a,b>0\) and \(x\in\mathbb{R}\),_  
1. \({}_{1}\mathrm{F}_{1}(a;b;x)\in\mathbb{R}\)\({}_{1}\)F\({}_{1}(a;b;x)>0\);  
\({}_{3}\)F\({}_{1}(a;b;x)\) is strictly increasing in \(x\);  
\(4\). _For \(x>0\), \({}_{1}\)F\({}_{1}(a;b;x)>1\)._  
Proof.: (1) This follows immediately from the power series expansion in Equation (28) and that \(a\), \(b\), and \(x\) are real numbers and \(b\) is positive. Since the real numbers are closed under arithmetic operations (\(b>0\) excludes division by zero) and are complete, the power series converges to a real number.  
(2) It is given in [30] that for \(a,b>0\), \({}_{1}\)F\({}_{1}(a;b;x)\) has no real zeros. Again, from the power series, it is seen that \({}_{1}\)F\({}_{1}(a;b;0)=1\). Hence, \({}_{1}\)F\({}_{1}(a;b;x)\) must remain positive for all real values of \(x\).  
(3) By Equation (31),  
\[\frac{d}{dx}\,{}_{1}\)F\({}_{1}(a;b;x)=\frac{a}{b}\,{}_{1}\)F\({}_{1}(a+1,b+1,x)\]  
and since \(a+1>a>0\) and \(b+1>b>0\), part 2 that was just proven can be applied using \(a+1\) and \(b+1\) to conclude  
\[\frac{d}{dx}\,{}_{1}\)F\({}_{1}(a;b;x)=\frac{a}{b}\,{}_{1}\)F\({}_{1}(a+1,b+1,x)>0.\]  
Hence, \({}_{1}\)F\({}_{1}(a;b;x)\) is strictly increasing in \(x\).  
(4) For \(0<x\), \(1={}_{1}\)F\({}_{1}(a;b;0)<{}_{1}\)F\({}_{1}(a;b;x)\) since it is strictly increasing in \(x\). Hence, the result follows.