Airy T bar integral. bigairydelta p_ : Pi 4 2 p^3 3. smallairydelta p_ : 3^ 2 3 Gamma 4 3 Gamma 2 3 p. smallairypade p_ : c p b p^3. a Pi 4 Π 4.
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1 Airy T bar integral In[110]:= In[111]:= In[112]:= In[113]:= bigairydelta p_ : Pi 4 2 p^3 3 smallairydelta p_ : 3^ 2 3 Gamma 4 3 Gamma 2 3 p bigairypade p_ : c p b p^3 b c a p^4 1 c a p smallairypade p_ : c p b p^3 In[114]:= Out[114]= a Pi 4 Π 4 In[115]:= Out[115]= In[116]:= Out[116]= In[117]:= In[118]:= In[119]:= b c 3^ 2 3 Gamma 4 3 Gamma Gamma 4 3 Gamma 2 3 airypade p_ : h p bigairypade p 1 h p smallairypade p h p_ : Tanh s p s 1.5 Out[119]= 1.5 In[120]:= Plot airypade p, smallairydelta p, bigairydelta p, p, 0, Out[120]= In[121]:= In[122]:= uint r_, z_ : NIntegrate Sqrt 1 u^2 Cos 2 z r u 2 airypade r u, u, 0, 1 uint 1, 1 Out[122]=
2 2 airytbar0.nb In[123]:= Plot uint r, 1, r, 0, Out[123]= 0.4 In[124]:= Plot uint r, 1, r, 1, Out[124]= In[125]:= In[126]:= s 2 Out[126]= 2 In[127]:= Plot uint r, 1, r, 0, Out[127]= 0.4
3 airytbar0.nb 3 In[128]:= Plot uint r, 1, r, 1, Out[128]= In[129]:= s 1 Out[129]= 1 In[130]:= Plot uint r, 1, r, 0, Out[130]= 0.4 In[131]:= s 10 Out[131]= 10 In[132]:= Plot uint r, 1, r, 0, Out[132]= 0.4
4 4 airytbar0.nb In[133]:= s 0.1 Out[133]= 0.1 In[134]:= Plot uint r, 1, r, 0, Out[134]= 0.4 In[135]:= s 1.5 Out[135]= 1.5 In[136]:= In[137]:= tbar0 z_ : N Pi^ 3 NIntegrate r uint r, z, r, 0, Infinity tbar0 1 Out[137]= NIntegrate::ncvb : NIntegrate failed to converge to prescribed accuracy after 9 recursive bisections in r near r NIntegrate obtained and for the integral and error estimates
5 airytbar0.nb 5 In[138]:= tbar0 0.1 Out[138]= 0. 1 u 2 Cos r u 2 Times 3 Times 5 1 Times 2 1 u 2 Cos r u 2 Times 3 Times 5 1 Times 2 1 u 2 Cos r u 2 Times 3 Times 5 1 Times 2 NIntegrate::slwcon : Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small. NIntegrate::ncvb : NIntegrate failed to converge to prescribed accuracy after 9 recursive bisections in r near r NIntegrate obtained and for the integral and error estimates.
6 6 airytbar0.nb In[139]:= tbar0 5 1 u 2 Cos 10 r u 2 Times 3 Times 5 1 Times 2 1 u 2 Cos 10 r u 2 Times 3 Times 5 1 Times 2 1 u 2 Cos 10 r u 2 Times 3 Times 5 1 Times 2 NIntegrate::ncvb : NIntegrate failed to converge to prescribed accuracy after 9 recursive bisections in r near r NIntegrate obtained and for the integral and error estimates. Out[139]= In[140]:= riesztbar0 z_, lambda_ : N Pi^ 3 1 z lambda NIntegrate r uint r, z, r, 0, lambda
7 airytbar0.nb 7 In[141]:= riesztbar0 1, 10 Out[141]= NIntegrate::slwcon : Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small In[142]:= tbar0 z_ : N Pi^ 3 NIntegrate r uint r, z, r, 0.5, 3
8 8 airytbar0.nb In[143]:= tbar0 1 Out[143]= In[144]:= uint 0.5, 1 Out[144]= In[145]:= uint 0, 1 Out[145]=
9 airytbar0.nb 9 In[146]:= NIntegrate uint r, 1, r, 0, 2 Out[146]= In[147]:= Plot uint r, 1, r, 0, Out[147]= 0.4
10 10 airytbar0.nb In[148]:= NIntegrate uint r, 1, r, 0.5, 0.7 Out[148]= In[149]:= NIntegrate uint r, 1, r, 0.5, 0.55 Out[149]=
11 airytbar0.nb 11 In[150]:= Out[150]= 1.5 NIntegrate x 1, x, 0, 1 In[151]:= NIntegrate NIntegrate x y, x, 0, 1, y, 1, 2 Out[151]= 2. x y has evaluated to non numerical values for all sampling points in the region with boundaries 0, 1. In[152]:= NIntegrate Integrate x y, x, 0, 1, y, 1, 2 Out[152]= 2. In[153]:= In[154]:= uint r_, z_ : Integrate Sqrt 1 u^2 Cos 2 z r u 2 airypade r u, u, 0, 1 tbar0 1 Out[154]= 1 r 1 u 2 Cos 2 r u 2 Times 2 Times 4 u 0 has evaluated to non numerical values for all sampling points in the region with boundaries 0.5, NIntegrate r uint r, 1, r, 0.5, 3 In[155]:= NIntegrate x y, x, 0, 1, y, 1, 2 Out[155]= 2. In[156]:= In[157]:= tbar z_ : NIntegrate r Sqrt 1 u^2 Cos 2 z r u 2 airypade r u, u, 0, 1, r, 0, Infinity tbar 1 NIntegrate::slwcon : Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small. NIntegrate::ncvb : NIntegrate failed to converge to prescribed accuracy after 18 recursive bisections in r near u, r , NIntegrate obtained *^209 and *^210 for the integral and error estimates. Out[157]= In[158]:= tbar 0.1 NIntegrate::slwcon : Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small. NIntegrate::ncvb : NIntegrate failed to converge to prescribed accuracy after 18 recursive bisections in r near u, r , NIntegrate obtained *^209 and *^210 for the integral and error estimates. Out[158]=
12 12 airytbar0.nb In[159]:= tbar 10 NIntegrate::slwcon : Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small. NIntegrate::ncvb : NIntegrate failed to converge to prescribed accuracy after 18 recursive bisections in r near u, r , NIntegrate obtained *^209 and *^210 for the integral and error estimates. Out[159]= In[160]:= s Out[160]= 1.5 In[161]:= s Out[161]= 1.5 In[162]:= uint r, z Out[162]= u 2 Cos 2 r u z 2 2 r 3 u r u Gamma Gamma Tanh r u 2 2 r 3 u r u Gamma 4 8 r 4 u 4 Gamma Tanh r u Gamma Π Gamma r u Gamma Π Gamma 2 3 u In[163]:=? uint Global uint uint r_, z_ : u 2 Cos 2 z r u 2 airypade r u u In[164]:= ruint u_, r_, z_ : Pi^ 3 r 1 u 2 Cos 2 z r u 2 airypade r u
13 airytbar0.nb In[165]:= r, - 1D, 8r, 0, 10<, 8u, 0, 1<D Out[165]= In[166]:= 0.0 Plot3D@ruint@u, r, - 1D, 8r, 1, 3<, 8u, 0, 1<D Out[166]=
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