# Approximations Involving Exponential Functions

I am reading a text and I am curious to know how certain approximations were reached.

The first function approximations is: $$1- \frac{1}{2p}((1+p)e^{\frac{-y}{x(1+p)}} - (1-p)e^{\frac{-y}{x(1-p)}}) \approx \frac{y^2}{2x^2 (1-p^2)}$$

,when $y \ll x$. Note that I tried using the approximation $e^x \approx 1+x$, when x is small, but all I got was the conclusion that $1- \frac{1}{2p}((1+p)e^{\frac{-y}{x(1+p)}} - (1-p)e^{\frac{-y}{x(1-p)}}) \approx 0$.

The second function approximation is: $$1-e^{\frac{-y}{x}}(1-Q(a,b)+Q(b,a)) \approx \frac{y^2}{x^2 (1-p^2)}$$

,when $y \ll x$, where $Q(a,b) = \int_b^\infty e^{-\frac12 (a^2 + u^2)} I_0(au) u \, du$, $b = \sqrt{\frac{2y}{x(1-p^2)}}$, $a = bp$, $I_0$ is a modified Bessel Function of the first kind. It is also a known fact that $$Q(b,0) = 1$$ and $$Q(0,b) = e^{-\frac{b^2}{2}}$$ I have tried to assume $$a = 0$$, since $a = bp$ and $b$ is small, $p$ is a number between 0 and 1. However, it is not clear how $(1-p^2)$ is in the denominator and not $(1-p^2)^2$, which would be closer to the traditional $e^x$ approximation.

Any hints on how these approximations were derived would be appreciated. Thanks.

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A common approximation for square root is $\sqrt{x+a} = \sqrt{x} + a \frac12 x^{-\frac12} + \frac{a^2}{2} \frac{-1}{4} x^{\frac{-3}{2}}$, but it is not clear what the values of $x,a$ should be. Also the Q functions must "pass" through, since the $(1-p^2)$ term passes through to the end. – jrand Apr 3 '11 at 23:27
Also, following Ross Millikan's answer, the approximation is probably of the form $1-(1-\frac{y}{x}+\frac{y^2}{x^2})(f(1-p^2))$, since the square terms must carry through. – jrand Apr 3 '11 at 23:32
For reference, could you please post a link to the "text" you were reading? – J. M. Apr 7 '11 at 5:31
Yes, the text is "Mobile Microwave Communications" by Jakes. The context is calculating the cumulative probability distribution of the average signal to noise ratio in a system with multiple copies of the same signal, when the signals are correlated. – jrand Apr 7 '11 at 7:38
Ah, I see why you need Marcum's function then... – J. M. Apr 7 '11 at 7:40

For the first one, you need to keep one more term in the expansion. $e^x \approx 1+x+\frac{x^2}{2!}$. When the first terms cancel, it is time for one more. That is how the squares appeared.

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Thanks. I tried this and it worked. – jrand Apr 3 '11 at 22:57

For the second one, use the following approximations, valid when a,b,x are small: $$Q(a,b) \approx 1 - \frac{b^2}{2} + \frac{a^2 b^2}{4} + \frac{b^4}{8}$$

$$e^x \approx 1 + x + \frac{x^2}{2!}$$

Then after the substitution has been made, the result will be: $$1 - (A)$$

A has 15 terms, since it is the result of multiplying the 3 terms from the exponential approximation by (1+4+4 = 9) minus (2 from Q terms cancelling) minus (2 from 1, -1 cancelling). As an example of 2 of the 15 terms in A, $\frac{-y}{x} \frac{p^2}{(1-p^2)}$ and a higher term one: $\frac{-y^4}{4x^4 (1-p^2)^2}$.

Then after all 15 terms in A have been reached, eliminate all the terms containing $y^3$. I believe the reason is since y is much smaller than x, ($\frac{y}{x})^3$ is even smaller. Terms will cancel out and the desired approximation will be reached.

For reference, the approximation is defined on page 473, equation (10-10-10) of "Communication Systems and Techniques" by Mischa Schwartz.

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