The following are based on a recent thread.

I found it good to collect the results.

(Version 5.2 is used)

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When Integrate returns conditional results (that is If[...]

structures)

it does not mean necessary that for the other values of the

parameter(s)

there is divergence or no closed form solution.

Below there is a nice example

In[10]:=

Integrate[E^(-x^2 - x*y)/Sqrt[x], {x, 0, Infinity}]

Out[10]=

If[Re[y] < 0, (E^(y^2/8)*Pi*Sqrt[-y]*(BesselI[-(1/4), y^2/8] +

BesselI[1/4, y^2/8]))/(2*Sqrt[2]),

Integrate[1/(E^(x*(x + y))*Sqrt[x]), {x, 0, Infinity}, Assumptions -

At first someone might think that for Re[y]>0 there is no convergence

or closed form solution. However it is just that the integrator found

a result

which is valid for Re[y]<0. Nothing more, nothing less.

Specifying assumptions shows the situation

In[11]:=

(Integrate[E^(-x^2 - x*y)/Sqrt[x], {x, 0, Infinity}, Assumptions ->

#1[Re[y], 0]] & ) /@ {Greater, Less}

Out[11]=

{(1/2)*E^(y^2/8)*Sqrt[y]*BesselK[1/4, y^2/8], (E^(y^2/8)*Pi*Sqrt[-

y]*(BesselI[-(1/4), y^2/8] + BesselI[1/4, y^2/8]))/

(2*Sqrt[2])}

A nice application of PiecewiseExpand is given below

In[11]:=

Integrate[E^(-x^2 - x*y)/Sqrt[x], {x, 0, Infinity}]

PiecewiseExpand[%]

Out[11]=

If[Re[y] < 0, (E^(y^2/8)*Pi*Sqrt[-y]*(BesselI[-(1/4), y^2/8] +

BesselI[1/4, y^2/8]))/(2*Sqrt[2]),

Integrate[1/(E^(x*(x + y))*Sqrt[x]), {x, 0, Infinity}, Assumptions -

Out[12]=

Piecewise[{{(E^(y^2/8)*Pi*Sqrt[-y]*(BesselI[-(1/4), y^2/8] +

BesselI[1/4, y^2/8]))/(2*Sqrt[2]), Re[y] <= 0}},

(1/2)*E^(y^2/8)*Sqrt[y]*BesselK[1/4, y^2/8]]

As another example consider

In[17]:=

Integrate[(1 + k*Sin[a]^2)^(1/2), {a, 0, 2*Pi}, Assumptions -> k

Reals]

PiecewiseExpand[%]

Out[17]=

If[k >= -1, 4*EllipticE[-k], Integrate[Sqrt[1 + k*Sin[a]^2], {a, 0,

2*Pi}, Assumptions -> k < -1]]

Out[18]=

4*EllipticE[-k]

which shows that although the If structure would indicate that there

is closed form

solution valid only for k>=-1, application of PiecewiseExpand shows

that the closed

form result is valid for all real k.

Dimitris

1. Conditional Statement with ANDs

Hi all,

I have the following query:

The two and clauses in the subquery need to be conditional.(if possible)

Only included in the select if @current_hazards = True Or @areas IS NOT NULL

If @current_hazards = False AND @areas IS NULL then both add statements are

to be ignored???

If this is possible i would appreciate any insight on how to achieve this...

Cheers,

Adam

SELECT

oce_rep_areas.oce_rep_area_id, oce_rep_areas.name

FROM

oce_rep_areas

WHERE

oce_rep_areas.oce_rep_area_id IN (

SELECT

parent

FROM

oce_rep_areas

INNER JOIN

oce_rep_items

ON

oce_rep_areas.oce_rep_area_id = oce_rep_items.oce_rep_area_id

WHERE

oce_rep_items.oce_report_id = @oceid

AND ///to bo conditional : ie add only if @hazards = True

oce_rep_items.current_hazard = True

AND //to be conditional : ie only include when @areas is not null

oce_rep_areas.oce_rep_area_id IN(@areas))

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