Temperature-Humidity Relationship - Revision history

Sharon Honecker: /* Conditional Reliability Function */ changed R(T,t,V,U) to R((t|T),V,U)

2017-02-08T21:12:29Z

Conditional Reliability Function: changed R(T,t,V,U) to R((t|T),V,U)

← Older revision		Revision as of 21:12, 8 February 2017
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	The T-H Weibull conditional reliability function at a specified stress level is given by:		The T-H Weibull conditional reliability function at a specified stress level is given by:

	::<math>R(T,t,V,U)=\frac{R(T+t,V,U)}{R(T,V,U)}=\frac{{{e}^{-{{\left( \tfrac{T+t}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}} \right)}^{\beta }}}}}{{{e}^{-{{\left( \tfrac{T}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}} \right)}^{\beta }}}}}\,\!</math>		::<math>R((t\|T),V,U)=\frac{R(T+t,V,U)}{R(T,V,U)}=\frac{{{e}^{-{{\left( \tfrac{T+t}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}} \right)}^{\beta }}}}}{{{e}^{-{{\left( \tfrac{T}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}} \right)}^{\beta }}}}}\,\!</math>

	or:		or:

	::<math>R(T,t,V,U)={{e}^{-\left[ {{\left( \tfrac{T+t}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}} \right)}^{\beta }}-{{\left( \tfrac{T}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}} \right)}^{\beta }} \right]}}\,\!</math>		::<math>R((t\|T),V,U)={{e}^{-\left[ {{\left( \tfrac{T+t}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}} \right)}^{\beta }}-{{\left( \tfrac{T}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}} \right)}^{\beta }} \right]}}\,\!</math>

	===Reliable Life===		===Reliable Life===

Sharon Honecker: /* Conditional Reliability */ changed R(T,t,V,U) to R((t|T),V,U)

2017-02-08T21:11:56Z

Conditional Reliability: changed R(T,t,V,U) to R((t|T),V,U)

← Older revision		Revision as of 21:11, 8 February 2017
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	The conditional reliability function for the T-H exponential model is given by:		The conditional reliability function for the T-H exponential model is given by:

	::<math>R(T,t,V,U)=\frac{R(T+t,V,U)}{R(T,V,U)}=\frac{{{e}^{-\lambda (T+t)}}}{{{e}^{-\lambda T}}}={{e}^{-\tfrac{t}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}}}\,\!</math>		::<math>R((t\|T),V,U)=\frac{R(T+t,V,U)}{R(T,V,U)}=\frac{{{e}^{-\lambda (T+t)}}}{{{e}^{-\lambda T}}}={{e}^{-\tfrac{t}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}}}\,\!</math>

	===Reliable Life===		===Reliable Life===

Lisa Hacker at 00:00, 24 June 2015

2015-06-24T00:00:29Z

← Older revision		Revision as of 00:00, 24 June 2015
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	& {{T}_{L}}= & {{e}^{T_{L}^{\prime }}}\text{ (Lower bound)}		& {{T}_{L}}= & {{e}^{T_{L}^{\prime }}}\text{ (Lower bound)}
	\end{align}\,\!</math>		\end{align}\,\!</math>

	~~[[Category: Completed Theoretical Review]]~~

Richard House: /* A look at the Parameters \phi \,\! and b\,\! */

2012-09-21T22:30:21Z

A look at the Parameters \phi \,\! and b\,\!

← Older revision		Revision as of 22:30, 21 September 2012
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	Note that the Life vs. Stress plots are plotted on a log-reciprocal scale. Also note that the points shown in these plots represent the life characteristics at the test stress levels (the data set was fitted to a Weibull distribution, thus the points represent the scale parameter, <math>\eta )\,\!</math>. For example, the points shown in the first figure represent <math>\eta \,\!</math> at each of the test temperature levels (two temperature levels were considered in this test).		Note that the Life vs. Stress plots are plotted on a log-reciprocal scale. Also note that the points shown in these plots represent the life characteristics at the test stress levels (the data set was fitted to a Weibull distribution, thus the points represent the scale parameter, <math>\eta )\,\!</math>. For example, the points shown in the first figure represent <math>\eta \,\!</math> at each of the test temperature levels (two temperature levels were considered in this test).

	===A look at the Parameters ~~<math>\phi \,\!</math>~~ and ~~<math>~~b~~\,\!</math>~~===		===A look at the Parameters Phi and b===
	Depending on which stress type is kept constant, it can be seen from the linearized T-H relationship that either the parameter <math>\phi \,\!</math> or the parameter <math>b\,\!</math> is the slope of the resulting line. If, for example, the humidity is kept constant then <math>\phi \,\!</math> is the slope of the life line in a Life vs. Temperature plot. The steeper the slope, the greater the dependency of product life to the temperature. In other words, <math>\phi \,\!</math> is a measure of the effect that temperature has on the life, and <math>b\,\!</math> is a measure of the effect that relative humidity has on the life. The larger the value of <math>\phi ,\,\!</math> the higher the dependency of the life on the temperature. Similarly, the larger the value of <math>b,\,\!</math> the higher the dependency of the life on the humidity.		Depending on which stress type is kept constant, it can be seen from the linearized T-H relationship that either the parameter <math>\phi \,\!</math> or the parameter <math>b\,\!</math> is the slope of the resulting line. If, for example, the humidity is kept constant then <math>\phi \,\!</math> is the slope of the life line in a Life vs. Temperature plot. The steeper the slope, the greater the dependency of product life to the temperature. In other words, <math>\phi \,\!</math> is a measure of the effect that temperature has on the life, and <math>b\,\!</math> is a measure of the effect that relative humidity has on the life. The larger the value of <math>\phi ,\,\!</math> the higher the dependency of the life on the temperature. Similarly, the larger the value of <math>b,\,\!</math> the higher the dependency of the life on the humidity.

	===T-H Data===		===T-H Data===

Richard House at 22:24, 21 September 2012

2012-09-21T22:24:39Z

Show changes

Richard House at 03:03, 30 August 2012

2012-08-30T03:03:11Z

← Older revision		Revision as of 03:03, 30 August 2012
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	=T-H Exponential=		=T-H Exponential=
	By setting <math>m=L(U,V)\,\!</math> in the exponential ~~<math>~~pdf~~</math>~~ we can obtain the T-H exponential ~~<math>~~pdf~~</math>~~:		By setting <math>m=L(U,V)\,\!</math> in the exponential ''pdf'' we can obtain the T-H exponential ''pdf'':

	::<math>f(t,V,U)=\frac{1}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}\cdot {{e}^{-\tfrac{t}{A}\cdot {{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}}}</math>		::<math>f(t,V,U)=\frac{1}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}\cdot {{e}^{-\tfrac{t}{A}\cdot {{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}}}</math>
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	::<math>\overline{T}=\int_{0}^{\infty }t\cdot f(t,V,U)dt</math>		::<math>\overline{T}=\int_{0}^{\infty }t\cdot f(t,V,U)dt</math>

	Substituting the T-H exponential ~~<math>~~pdf~~</math>~~ equation yields:		Substituting the T-H exponential ''pdf'' equation yields:

	::<math>\begin{align}		::<math>\begin{align}
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	=T-H Weibull=<!-- THIS SECTION HEADER IS LINKED TO: T-H_Example. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK. -->		=T-H Weibull=<!-- THIS SECTION HEADER IS LINKED TO: T-H_Example. IF YOU RENAME THE SECTION, YOU MUST UPDATE THE LINK. -->
	By setting <math>\eta =L(U,V)\,\!</math> in the Weibull ~~<math>~~pdf~~</math>~~, the T--H Weibull model's ~~<math>~~pdf~~</math>~~ is given by:		By setting <math>\eta =L(U,V)\,\!</math> in the Weibull ''pdf'', the T--H Weibull model's ''pdf'' is given by:

	::<math>f(t,\,V,\,U)=\frac{\beta }{A}{{e}^{-\left( \tfrac{\varphi }{V}+\tfrac{b}{U} \right)}}{{\left( \frac{t}{A}{{e}^{-\left( \tfrac{\varphi }{V}+\tfrac{b}{U} \right)}} \right)}^{\beta -1}}{{e}^{-{{\left( \tfrac{t}{A}{{e}^{-\left( \tfrac{\varphi }{V}+\tfrac{b}{U} \right)}} \right)}^{\beta }}}}</math>		::<math>f(t,\,V,\,U)=\frac{\beta }{A}{{e}^{-\left( \tfrac{\varphi }{V}+\tfrac{b}{U} \right)}}{{\left( \frac{t}{A}{{e}^{-\left( \tfrac{\varphi }{V}+\tfrac{b}{U} \right)}} \right)}^{\beta -1}}{{e}^{-{{\left( \tfrac{t}{A}{{e}^{-\left( \tfrac{\varphi }{V}+\tfrac{b}{U} \right)}} \right)}^{\beta }}}}</math>
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	=T-H Lognormal=		=T-H Lognormal=
	The ~~<math>~~pdf~~</math>~~ of the lognormal distribution is given by:		The ''pdf'' of the lognormal distribution is given by:

	::<math>f(T)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\overline{{{T}'}}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}</math>		::<math>f(T)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\overline{{{T}'}}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}</math>
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	::<math>\breve{T}={{e}^{{{\overline{T}}^{\prime }}}}</math>		::<math>\breve{T}={{e}^{{{\overline{T}}^{\prime }}}}</math>

	The T-H lognormal model ~~<math>~~pdf~~</math>~~ can be obtained first by setting <math>\breve{T} =L(V,U)\,\!</math>. <br>		The T-H lognormal model ''pdf'' can be obtained first by setting <math>\breve{T} =L(V,U)\,\!</math>. <br>
	Therefore:		Therefore:

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	::<math>{{\overline{T}}^{\prime }}=\ln (A)+\frac{\phi }{V}+\frac{b}{U}.</math>		::<math>{{\overline{T}}^{\prime }}=\ln (A)+\frac{\phi }{V}+\frac{b}{U}.</math>

	Substituting the above equation into the lognormal ~~<math>~~pdf~~</math>~~ yields the T-H lognormal model ~~<math>~~pdf~~</math>~~ or:		Substituting the above equation into the lognormal ''pdf'' yields the T-H lognormal model ''pdf'' or:

	::<math>f(T,V,U)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\ln (A)-\tfrac{\phi }{V}-\tfrac{b}{U}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}</math>		::<math>f(T,V,U)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\ln (A)-\tfrac{\phi }{V}-\tfrac{b}{U}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}</math>

Richard House: /* Bounds on the Parameters */

2012-08-23T00:57:42Z

Bounds on the Parameters

← Older revision		Revision as of 00:57, 23 August 2012
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	{{\sigma }_{L}}=\ & \frac{{{\widehat{\sigma }}_{{{T}'}}}}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var({{\widehat{\sigma }}_{{{T}'}}})}}{{{\widehat{\sigma }}_{{{T}'}}}}}}}&\text{ (Lower bound)}		{{\sigma }_{L}}=\ & \frac{{{\widehat{\sigma }}_{{{T}'}}}}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var({{\widehat{\sigma }}_{{{T}'}}})}}{{{\widehat{\sigma }}_{{{T}'}}}}}}}&\text{ (Lower bound)}
	\end{align}</math>		\end{align}</math>
	~~<br>~~
	and:		and:

	~~<br>~~
	::<math>\begin{align}		::<math>\begin{align}
	{{A}_{U}}=\ & \widehat{A}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{A})}}{\widehat{A}}}}&\text{ (Upper bound)} \\		{{A}_{U}}=\ & \widehat{A}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{A})}}{\widehat{A}}}}&\text{ (Upper bound)} \\
	{{A}_{L}}=\ & \frac{\widehat{A}}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{A})}}{\widehat{A}}}}}&\text{ (Lower bound)}		{{A}_{L}}=\ & \frac{\widehat{A}}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{A})}}{\widehat{A}}}}}&\text{ (Lower bound)}
	\end{align}</math>		\end{align}</math>
	~~<br>~~
	The lower and upper bounds on <math>\phi \,\!</math> and <math>b\,\!</math> are estimated from:		The lower and upper bounds on <math>\phi \,\!</math> and <math>b\,\!</math> are estimated from:

	~~<br>~~
	::<math>\begin{align}		::<math>\begin{align}
	& {{\phi }_{U}}= & \widehat{\phi }+{{K}_{\alpha }}\sqrt{Var(\widehat{\phi })}\text{ (Upper bound)} \\		& {{\phi }_{U}}= & \widehat{\phi }+{{K}_{\alpha }}\sqrt{Var(\widehat{\phi })}\text{ (Upper bound)} \\
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	\end{align}</math>		\end{align}</math>

	~~<br>~~
	and:		and:

	~~<br>~~
	::<math>\begin{align}		::<math>\begin{align}
	& {{b}_{U}}= & \widehat{b}+{{K}_{\alpha }}\sqrt{Var(\widehat{b})}\text{ (Upper bound)} \\		& {{b}_{U}}= & \widehat{b}+{{K}_{\alpha }}\sqrt{Var(\widehat{b})}\text{ (Upper bound)} \\
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	\end{align}</math>		\end{align}</math>

	~~<br>~~
	The variances and covariances of <math>A\,\!</math> , <math>\phi ,\,\!</math> <math>b,\,\!</math> and <math>{{\sigma }_{{{T}'}}}\,\!</math> are estimated from the local Fisher matrix (evaluated at <math>\widehat{A}</math> , <math>\widehat{\phi },</math> <math>\widehat{b}</math> , <math>{{\widehat{\sigma }}_{{{T}'}}}),</math> as follows:		The variances and covariances of <math>A\,\!</math> , <math>\phi ,\,\!</math> <math>b,\,\!</math> and <math>{{\sigma }_{{{T}'}}}\,\!</math> are estimated from the local Fisher matrix (evaluated at <math>\widehat{A}</math> , <math>\widehat{\phi },</math> <math>\widehat{b}</math> , <math>{{\widehat{\sigma }}_{{{T}'}}}),</math> as follows:

	~~<br>~~
	::<math>\left( \begin{matrix}		::<math>\left( \begin{matrix}
	Var\left( {{\widehat{\sigma }}_{{{T}'}}} \right) & Cov\left( \widehat{A},{{\widehat{\sigma }}_{{{T}'}}} \right) & Cov\left( \widehat{\phi },{{\widehat{\sigma }}_{{{T}'}}} \right) & Cov\left( \widehat{b},{{\widehat{\sigma }}_{{{T}'}}} \right) \\		Var\left( {{\widehat{\sigma }}_{{{T}'}}} \right) & Cov\left( \widehat{A},{{\widehat{\sigma }}_{{{T}'}}} \right) & Cov\left( \widehat{\phi },{{\widehat{\sigma }}_{{{T}'}}} \right) & Cov\left( \widehat{b},{{\widehat{\sigma }}_{{{T}'}}} \right) \\
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	\end{matrix} \right)={{F}^{-1}}</math>		\end{matrix} \right)={{F}^{-1}}</math>

	~~<br>~~
	where:		where:

	~~<br>~~
	::<math>{{F}^{-1}}={{\left( \begin{matrix}		::<math>{{F}^{-1}}={{\left( \begin{matrix}
	-\tfrac{{{\partial }^{2}}\Lambda }{\partial \sigma _{{{T}'}}^{2}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_{{{T}'}}}\partial A} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_{{{T}'}}}\partial \phi } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_{{{T}'}}}\partial b} \\		-\tfrac{{{\partial }^{2}}\Lambda }{\partial \sigma _{{{T}'}}^{2}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_{{{T}'}}}\partial A} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_{{{T}'}}}\partial \phi } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_{{{T}'}}}\partial b} \\
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	-\tfrac{{{\partial }^{2}}\Lambda }{\partial b\partial {{\sigma }_{{{T}'}}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial b\partial A} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial b\partial \phi } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{b}^{2}}} \\		-\tfrac{{{\partial }^{2}}\Lambda }{\partial b\partial {{\sigma }_{{{T}'}}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial b\partial A} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial b\partial \phi } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{b}^{2}}} \\
	\end{matrix} \right)}^{-1}}</math>		\end{matrix} \right)}^{-1}}</math>

	~~<br>~~

	===Bounds on Reliability===		===Bounds on Reliability===

Richard House: /* Bounds on Reliability */

2012-08-23T00:56:41Z

Bounds on Reliability

← Older revision		Revision as of 00:56, 23 August 2012
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	Let <math>\widehat{z}(t,V,U;A,\phi ,b,{{\sigma }_{T}})=\tfrac{t-\ln (\widehat{A})-\tfrac{\widehat{\phi }}{V}-\tfrac{\widehat{b}}{U}}{{{\widehat{\sigma }}_{{{T}'}}}},</math> then <math>\tfrac{d\widehat{z}}{dt}=\tfrac{1}{{{\widehat{\sigma }}_{{{T}'}}}}.</math>		Let <math>\widehat{z}(t,V,U;A,\phi ,b,{{\sigma }_{T}})=\tfrac{t-\ln (\widehat{A})-\tfrac{\widehat{\phi }}{V}-\tfrac{\widehat{b}}{U}}{{{\widehat{\sigma }}_{{{T}'}}}},</math> then <math>\tfrac{d\widehat{z}}{dt}=\tfrac{1}{{{\widehat{\sigma }}_{{{T}'}}}}.</math>
	For <math>t={T}'</math> , <math>\widehat{z}=\tfrac{{T}'-\ln (\widehat{A})-\tfrac{\widehat{\phi }}{V}-\tfrac{\widehat{b}}{U}}{{{\widehat{\sigma }}_{{{T}'}}}}</math> , and for <math>t=\infty ,\,\!</math> <math>\widehat{z}=\infty .</math> The above equation then becomes:		For <math>t={T}'\,\!</math> , <math>\widehat{z}=\tfrac{{T}'-\ln (\widehat{A})-\tfrac{\widehat{\phi }}{V}-\tfrac{\widehat{b}}{U}}{{{\widehat{\sigma }}_{{{T}'}}}}</math> , and for <math>t=\infty ,\,\!</math> <math>\widehat{z}=\infty .</math> The above equation then becomes:

	::<math>R(\widehat{z})=\int_{\widehat{z}({T}',V,U)}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz</math>		::<math>R(\widehat{z})=\int_{\widehat{z}({T}',V,U)}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz</math>

Richard House: /* Bounds on Reliability */

2012-08-23T00:56:24Z

Bounds on Reliability

← Older revision		Revision as of 00:56, 23 August 2012
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	Let <math>\widehat{z}(t,V,U;A,\phi ,b,{{\sigma }_{T}})=\tfrac{t-\ln (\widehat{A})-\tfrac{\widehat{\phi }}{V}-\tfrac{\widehat{b}}{U}}{{{\widehat{\sigma }}_{{{T}'}}}},</math> then <math>\tfrac{d\widehat{z}}{dt}=\tfrac{1}{{{\widehat{\sigma }}_{{{T}'}}}}.</math>		Let <math>\widehat{z}(t,V,U;A,\phi ,b,{{\sigma }_{T}})=\tfrac{t-\ln (\widehat{A})-\tfrac{\widehat{\phi }}{V}-\tfrac{\widehat{b}}{U}}{{{\widehat{\sigma }}_{{{T}'}}}},</math> then <math>\tfrac{d\widehat{z}}{dt}=\tfrac{1}{{{\widehat{\sigma }}_{{{T}'}}}}.</math>
	For <math>t={T}'</math> , <math>\widehat{z}=\tfrac{{T}'-\ln (\widehat{A})-\tfrac{\widehat{\phi }}{V}-\tfrac{\widehat{b}}{U}}{{{\widehat{\sigma }}_{{{T}'}}}}</math> , and for <math>t=\infty ,</math> <math>\widehat{z}=\infty .</math> The above equation then becomes:		For <math>t={T}'</math> , <math>\widehat{z}=\tfrac{{T}'-\ln (\widehat{A})-\tfrac{\widehat{\phi }}{V}-\tfrac{\widehat{b}}{U}}{{{\widehat{\sigma }}_{{{T}'}}}}</math> , and for <math>t=\infty ,\,\!</math> <math>\widehat{z}=\infty .</math> The above equation then becomes:

	::<math>R(\widehat{z})=\int_{\widehat{z}({T}',V,U)}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz</math>		::<math>R(\widehat{z})=\int_{\widehat{z}({T}',V,U)}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz</math>

Richard House: /* Bounds on Reliability */

2012-08-23T00:55:39Z

Bounds on Reliability

← Older revision		Revision as of 00:55, 23 August 2012
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	::<math>R(\widehat{z})=\int_{\widehat{z}({T}',V,U)}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz</math>		::<math>R(\widehat{z})=\int_{\widehat{z}({T}',V,U)}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz</math>

	The bounds on <math>z</math> are estimated from:		The bounds on <math>z\,\!</math> are estimated from:

	::<math>\begin{align}		::<math>\begin{align}