10.61

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
10.61.E1	${\displaystyle{\displaystyle\operatorname{ber}_{\nu}x+i\operatorname{bei}_{\nu}x=J_{\nu}\left(xe^{3\pi i/4}\right)}}$ \Kelvinber{\nu}@@{x}+i\Kelvinbei{\nu}@@{x} = \BesselJ{\nu}@{xe^{3\pi i/4}}	${\displaystyle{\displaystyle\Re(\nu+k+1)>0}}$	KelvinBer(nu, x)+ I*KelvinBei(nu, x) = BesselJ(nu, x*exp(3*Pi*I/4))	KelvinBer[\[Nu], x]+ I*KelvinBei[\[Nu], x] == BesselJ[\[Nu], x*Exp[3*Pi*I/4]]	Successful	Failure	Skip - symbolical successful subtest	Successful [Tested: 30]
10.61.E1	${\displaystyle{\displaystyle J_{\nu}\left(xe^{3\pi i/4}\right)=e^{\nu\pi i}J_{\nu}\left(xe^{-\pi i/4}\right)}}$ \BesselJ{\nu}@{xe^{3\pi i/4}} = e^{\nu\pi i}\BesselJ{\nu}@{xe^{-\pi i/4}}	${\displaystyle{\displaystyle\Re(\nu+k+1)>0}}$	BesselJ(nu, x*exp(3*Pi*I/4)) = exp(nu*Pi*I)*BesselJ(nu, x*exp(- Pi*I/4))	BesselJ[\[Nu], x*Exp[3*Pi*I/4]] == Exp[\[Nu]*Pi*I]*BesselJ[\[Nu], x*Exp[- Pi*I/4]]	Failure	Failure	Successful [Tested: 30]	Successful [Tested: 30]
10.61.E1	${\displaystyle{\displaystyle e^{\nu\pi i}J_{\nu}\left(xe^{-\pi i/4}\right)=e^{\nu\pi i/2}I_{\nu}\left(xe^{\pi i/4}\right)}}$ e^{\nu\pi i}\BesselJ{\nu}@{xe^{-\pi i/4}} = e^{\nu\pi i/2}\modBesselI{\nu}@{xe^{\pi i/4}}	${\displaystyle{\displaystyle\Re(\nu+k+1)>0}}$	exp(nu*Pi*I)*BesselJ(nu, x*exp(- Pi*I/4)) = exp(nu*Pi*I/2)*BesselI(nu, x*exp(Pi*I/4))	Exp[\[Nu]*Pi*I]*BesselJ[\[Nu], x*Exp[- Pi*I/4]] == Exp[\[Nu]*Pi*I/2]*BesselI[\[Nu], x*Exp[Pi*I/4]]	Failure	Failure	Successful [Tested: 30]	Successful [Tested: 30]
10.61.E1	${\displaystyle{\displaystyle e^{\nu\pi i/2}I_{\nu}\left(xe^{\pi i/4}\right)=e^{3\nu\pi i/2}I_{\nu}\left(xe^{-3\pi i/4}\right)}}$ e^{\nu\pi i/2}\modBesselI{\nu}@{xe^{\pi i/4}} = e^{3\nu\pi i/2}\modBesselI{\nu}@{xe^{-3\pi i/4}}	${\displaystyle{\displaystyle\Re(\nu+k+1)>0}}$	exp(nu*Pi*I/2)*BesselI(nu, x*exp(Pi*I/4)) = exp(3*nu*Pi*I/2)*BesselI(nu, x*exp(- 3*Pi*I/4))	Exp[\[Nu]*Pi*I/2]*BesselI[\[Nu], x*Exp[Pi*I/4]] == Exp[3*\[Nu]*Pi*I/2]*BesselI[\[Nu], x*Exp[- 3*Pi*I/4]]	Failure	Failure	Successful [Tested: 30]	Successful [Tested: 30]
10.61.E2	${\displaystyle{\displaystyle\operatorname{ker}_{\nu}x+i\operatorname{kei}_{\nu}x=e^{-\nu\pi i/2}K_{\nu}\left(xe^{\pi i/4}\right)}}$ \Kelvinker{\nu}@@{x}+i\Kelvinkei{\nu}@@{x} = e^{-\nu\pi i/2}\modBesselK{\nu}@{xe^{\pi i/4}}		KelvinKer(nu, x)+ I*KelvinKei(nu, x) = exp(- nu*Pi*I/2)*BesselK(nu, x*exp(Pi*I/4))	KelvinKer[\[Nu], x]+ I*KelvinKei[\[Nu], x] == Exp[- \[Nu]*Pi*I/2]*BesselK[\[Nu], x*Exp[Pi*I/4]]	Failure	Failure	Successful [Tested: 30]	Successful [Tested: 30]
10.61.E2	${\displaystyle{\displaystyle e^{-\nu\pi i/2}K_{\nu}\left(xe^{\pi i/4}\right)=\tfrac{1}{2}\pi i{H^{(1)}_{\nu}}\left(xe^{3\pi i/4}\right)}}$ e^{-\nu\pi i/2}\modBesselK{\nu}@{xe^{\pi i/4}} = \tfrac{1}{2}\pi i\HankelH{1}{\nu}@{xe^{3\pi i/4}}		exp(- nu*Pi*I/2)*BesselK(nu, x*exp(Pi*I/4)) = (1)/(2)*Pi*I*HankelH1(nu, x*exp(3*Pi*I/4))	Exp[- \[Nu]*Pi*I/2]*BesselK[\[Nu], x*Exp[Pi*I/4]] == Divide[1,2]*Pi*I*HankelH1[\[Nu], x*Exp[3*Pi*I/4]]	Failure	Failure	Successful [Tested: 30]	Successful [Tested: 30]
10.61.E2	${\displaystyle{\displaystyle\tfrac{1}{2}\pi i{H^{(1)}_{\nu}}\left(xe^{3\pi i/4}\right)=-\tfrac{1}{2}\pi ie^{-\nu\pi i}{H^{(2)}_{\nu}}\left(xe^{-\pi i/4}\right)}}$ \tfrac{1}{2}\pi i\HankelH{1}{\nu}@{xe^{3\pi i/4}} = -\tfrac{1}{2}\pi ie^{-\nu\pi i}\HankelH{2}{\nu}@{xe^{-\pi i/4}}		(1)/(2)*Pi*I*HankelH1(nu, x*exp(3*Pi*I/4)) = -(1)/(2)*Pi*I*exp(- nu*Pi*I)*HankelH2(nu, x*exp(- Pi*I/4))	Divide[1,2]*Pi*I*HankelH1[\[Nu], x*Exp[3*Pi*I/4]] == -Divide[1,2]*Pi*I*Exp[- \[Nu]*Pi*I]*HankelH2[\[Nu], x*Exp[- Pi*I/4]]	Failure	Failure	Successful [Tested: 30]	Successful [Tested: 30]
10.61.E3	${\displaystyle{\displaystyle x^{2}\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}x}^{2}}+x\frac{\mathrm{d}w}{\mathrm{d}x}-(ix^{2}+\nu^{2})w=0}}$ x^{2}\deriv[2]{w}{x}+x\deriv{w}{x}-(ix^{2}+\nu^{2})w = 0		(x)^(2)* diff(w, [x$(2)])+ x*diff(w, x)-(I*(x)^(2)+ (nu)^(2))*w = 0	(x)^(2)* D[w, {x, 2}]+ x*D[w, x]-(I*(x)^(2)+ \[Nu]^(2))*w == 0	Failure	Failure	Failed [300 / 300] Result: 1.125000000-2.948557160*I Test Values: {nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, x = 3/2} Result: .1249999997-1.216506352*I Test Values: {nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, x = 1/2} ... skip entries to safe data	Failed [300 / 300] Result: Complex[1.1249999999999996, -2.948557158514987] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[1.1249999999999996, -0.9485571585149869] Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.61.E4	${\displaystyle{\displaystyle x^{4}\frac{{\mathrm{d}}^{4}w}{{\mathrm{d}x}^{4}}+2x^{3}\frac{{\mathrm{d}}^{3}w}{{\mathrm{d}x}^{3}}-(1+2\nu^{2})\left(x^{2}\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}x}^{2}}-x\frac{\mathrm{d}w}{\mathrm{d}x}\right)+(\nu^{4}-4\nu^{2}+x^{4})w=0}}$ x^{4}\deriv[4]{w}{x}+2x^{3}\deriv[3]{w}{x}-(1+2\nu^{2})\left(x^{2}\deriv[2]{w}{x}-x\deriv{w}{x}\right)+(\nu^{4}-4\nu^{2}+x^{4})w = 0	${\displaystyle{\displaystyle w=\operatorname{ber}_{+\nu}x,w=\operatorname{ber}_{-\nu}x}}$	(x)^(4)* diff(w, [x$(4)])+ 2*(x)^(3)* diff(w, [x$(3)])-(1 + 2*(nu)^(2))*((x)^(2)* diff(w, [x$(2)])- x*diff(w, x))+((nu)^(4)- 4*(nu)^(2)+ (x)^(4))*w = 0	(x)^(4)* D[w, {x, 4}]+ 2*(x)^(3)* D[w, {x, 3}]-(1 + 2*\[Nu]^(2))*((x)^(2)* D[w, {x, 2}]- x*D[w, x])+(\[Nu]^(4)- 4*\[Nu]^(2)+ (x)^(4))*w == 0	Error	Failure	-	Skip - No test values generated
10.61#Ex1	${\displaystyle{\displaystyle\operatorname{ber}_{n}\left(-x\right)=(-1)^{n}\operatorname{ber}_{n}x}}$ \Kelvinber{n}@{-x} = (-1)^{n}\Kelvinber{n}@@{x}	${\displaystyle{\displaystyle\Re(n+k+1)>0}}$	KelvinBer(n, - x) = (- 1)^(n)* KelvinBer(n, x)	KelvinBer[n, - x] == (- 1)^(n)* KelvinBer[n, x]	Successful	Failure	-	Successful [Tested: 9]
10.61#Ex2	${\displaystyle{\displaystyle\operatorname{bei}_{n}\left(-x\right)=(-1)^{n}\operatorname{bei}_{n}x}}$ \Kelvinbei{n}@{-x} = (-1)^{n}\Kelvinbei{n}@@{x}		KelvinBei(n, - x) = (- 1)^(n)* KelvinBei(n, x)	KelvinBei[n, - x] == (- 1)^(n)* KelvinBei[n, x]	Successful	Failure	-	Successful [Tested: 9]
10.61#Ex3	${\displaystyle{\displaystyle\operatorname{ber}_{-\nu}x=\cos\left(\nu\pi\right)\operatorname{ber}_{\nu}x+\sin\left(\nu\pi\right)\operatorname{bei}_{\nu}x+(2/\pi)\sin\left(\nu\pi\right)\operatorname{ker}_{\nu}x}}$ \Kelvinber{-\nu}@@{x} = \cos@{\nu\pi}\Kelvinber{\nu}@@{x}+\sin@{\nu\pi}\Kelvinbei{\nu}@@{x}+(2/\pi)\sin@{\nu\pi}\Kelvinker{\nu}@@{x}	${\displaystyle{\displaystyle\Re((-\nu)+k+1)>0,\Re(\nu+k+1)>0}}$	KelvinBer(- nu, x) = cos(nu*Pi)*KelvinBer(nu, x)+ sin(nu*Pi)*KelvinBei(nu, x)+(2/Pi)*sin(nu*Pi)*KelvinKer(nu, x)	KelvinBer[- \[Nu], x] == Cos[\[Nu]*Pi]*KelvinBer[\[Nu], x]+ Sin[\[Nu]*Pi]*KelvinBei[\[Nu], x]+(2/Pi)*Sin[\[Nu]*Pi]*KelvinKer[\[Nu], x]	Failure	Failure	Successful [Tested: 30]	Successful [Tested: 30]
10.61#Ex4	${\displaystyle{\displaystyle\operatorname{bei}_{-\nu}x=-\sin\left(\nu\pi\right)\operatorname{ber}_{\nu}x+\cos\left(\nu\pi\right)\operatorname{bei}_{\nu}x+(2/\pi)\sin\left(\nu\pi\right)\operatorname{kei}_{\nu}x}}$ \Kelvinbei{-\nu}@@{x} = -\sin@{\nu\pi}\Kelvinber{\nu}@@{x}+\cos@{\nu\pi}\Kelvinbei{\nu}@@{x}+(2/\pi)\sin@{\nu\pi}\Kelvinkei{\nu}@@{x}	${\displaystyle{\displaystyle\Re(\nu+k+1)>0}}$	KelvinBei(- nu, x) = - sin(nu*Pi)*KelvinBer(nu, x)+ cos(nu*Pi)*KelvinBei(nu, x)+(2/Pi)*sin(nu*Pi)*KelvinKei(nu, x)	KelvinBei[- \[Nu], x] == - Sin[\[Nu]*Pi]*KelvinBer[\[Nu], x]+ Cos[\[Nu]*Pi]*KelvinBei[\[Nu], x]+(2/Pi)*Sin[\[Nu]*Pi]*KelvinKei[\[Nu], x]	Failure	Failure	Successful [Tested: 30]	Successful [Tested: 30]
10.61#Ex5	${\displaystyle{\displaystyle\operatorname{ker}_{-\nu}x=\cos\left(\nu\pi\right)\operatorname{ker}_{\nu}x-\sin\left(\nu\pi\right)\operatorname{kei}_{\nu}x}}$ \Kelvinker{-\nu}@@{x} = \cos@{\nu\pi}\Kelvinker{\nu}@@{x}-\sin@{\nu\pi}\Kelvinkei{\nu}@@{x}		KelvinKer(- nu, x) = cos(nu*Pi)*KelvinKer(nu, x)- sin(nu*Pi)*KelvinKei(nu, x)	KelvinKer[- \[Nu], x] == Cos[\[Nu]*Pi]*KelvinKer[\[Nu], x]- Sin[\[Nu]*Pi]*KelvinKei[\[Nu], x]	Successful	Failure	-	Successful [Tested: 30]
10.61#Ex6	${\displaystyle{\displaystyle\operatorname{kei}_{-\nu}x=\sin\left(\nu\pi\right)\operatorname{ker}_{\nu}x+\cos\left(\nu\pi\right)\operatorname{kei}_{\nu}x}}$ \Kelvinkei{-\nu}@@{x} = \sin@{\nu\pi}\Kelvinker{\nu}@@{x}+\cos@{\nu\pi}\Kelvinkei{\nu}@@{x}		KelvinKei(- nu, x) = sin(nu*Pi)*KelvinKer(nu, x)+ cos(nu*Pi)*KelvinKei(nu, x)	KelvinKei[- \[Nu], x] == Sin[\[Nu]*Pi]*KelvinKer[\[Nu], x]+ Cos[\[Nu]*Pi]*KelvinKei[\[Nu], x]	Successful	Failure	-	Successful [Tested: 30]
10.61#Ex7	${\displaystyle{\displaystyle\operatorname{ber}_{-n}x=(-1)^{n}\operatorname{ber}_{n}x,~{}{}\operatorname{bei}_{-n}x}}$ \Kelvinber{-n}@@{x} = (-1)^{n}\Kelvinber{n}@@{x},~{}\Kelvinbei{-n}@@{x}	${\displaystyle{\displaystyle\Re((-n)+k+1)>0,\Re(n+k+1)>0}}$	KelvinBer(- n, x) = (- 1)^(n)* KelvinBer(n, x); *KelvinBei(- n, x)	KelvinBer[- n, x] == (- 1)^(n)* KelvinBer[n, x] *KelvinBei[- n, x]	Error	Failure	-	Error
10.61#Ex7	${\displaystyle{\displaystyle(-1)^{n}\operatorname{ber}_{n}x,~{}{}\operatorname{bei}_{-n}x=(-1)^{n}\operatorname{bei}_{n}x}}$ (-1)^{n}\Kelvinber{n}@@{x},~{}\Kelvinbei{-n}@@{x} = (-1)^{n}\Kelvinbei{n}@@{x}	${\displaystyle{\displaystyle\Re((-n)+k+1)>0,\Re(n+k+1)>0}}$	(- 1)^(n)* KelvinBer(n, x),*KelvinBei(- n, x) = (- 1)^(n)* KelvinBei(n, x)	(- 1)^(n)* KelvinBer[n, x],*KelvinBei[- n, x] == (- 1)^(n)* KelvinBei[n, x]	Error	Failure	-	Error
10.61#Ex8	${\displaystyle{\displaystyle\operatorname{ker}_{-n}x=(-1)^{n}\operatorname{ker}_{n}x,~{}{}\operatorname{kei}_{-n}x}}$ \Kelvinker{-n}@@{x} = (-1)^{n}\Kelvinker{n}@@{x},~{}\Kelvinkei{-n}@@{x}		KelvinKer(- n, x) = (- 1)^(n)* KelvinKer(n, x); *KelvinKei(- n, x)	KelvinKer[- n, x] == (- 1)^(n)* KelvinKer[n, x] *KelvinKei[- n, x]	Error	Failure	-	Error
10.61#Ex8	${\displaystyle{\displaystyle(-1)^{n}\operatorname{ker}_{n}x,~{}{}\operatorname{kei}_{-n}x=(-1)^{n}\operatorname{kei}_{n}x}}$ (-1)^{n}\Kelvinker{n}@@{x},~{}\Kelvinkei{-n}@@{x} = (-1)^{n}\Kelvinkei{n}@@{x}		(- 1)^(n)* KelvinKer(n, x),*KelvinKei(- n, x) = (- 1)^(n)* KelvinKei(n, x)	(- 1)^(n)* KelvinKer[n, x],*KelvinKei[- n, x] == (- 1)^(n)* KelvinKei[n, x]	Error	Failure	-	Error
10.61#Ex9	${\displaystyle{\displaystyle\operatorname{ber}_{\frac{1}{2}}\left(x\sqrt{2}\right)=\frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}\left(e^{x}\cos\left(x+\frac{\pi}{8}\right)-e^{-x}\cos\left(x-\frac{\pi}{8}\right)\right)}}$ \Kelvinber{\frac{1}{2}}@{x\sqrt{2}} = \frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}\left(e^{x}\cos@{x+\frac{\pi}{8}}-e^{-x}\cos@{x-\frac{\pi}{8}}\right)	${\displaystyle{\displaystyle\Re((\frac{1}{2})+k+1)>0}}$	KelvinBer((1)/(2), x*sqrt(2)) = ((2)^(-(3)/(4)))/(sqrt(Pi*x))*(exp(x)*cos(x +(Pi)/(8))- exp(- x)*cos(x -(Pi)/(8)))	KelvinBer[Divide[1,2], x*Sqrt[2]] == Divide[(2)^(-Divide[3,4]),Sqrt[Pi*x]]*(Exp[x]*Cos[x +Divide[Pi,8]]- Exp[- x]*Cos[x -Divide[Pi,8]])	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
10.61#Ex10	${\displaystyle{\displaystyle\operatorname{bei}_{\frac{1}{2}}\left(x\sqrt{2}\right)=\frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}\left(e^{x}\sin\left(x+\frac{\pi}{8}\right)+\,e^{-x}\sin\left(x-\frac{\pi}{8}\right)\right)}}$ \Kelvinbei{\frac{1}{2}}@{x\sqrt{2}} = \frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}\left(e^{x}\sin@{x+\frac{\pi}{8}}+\,e^{-x}\sin@{x-\frac{\pi}{8}}\right)		KelvinBei((1)/(2), x*sqrt(2)) = ((2)^(-(3)/(4)))/(sqrt(Pi*x))*(exp(x)*sin(x +(Pi)/(8))+ exp(- x)*sin(x -(Pi)/(8)))	KelvinBei[Divide[1,2], x*Sqrt[2]] == Divide[(2)^(-Divide[3,4]),Sqrt[Pi*x]]*(Exp[x]*Sin[x +Divide[Pi,8]]+ Exp[- x]*Sin[x -Divide[Pi,8]])	Failure	Successful	Successful [Tested: 3]	Successful [Tested: 3]
10.61#Ex11	${\displaystyle{\displaystyle\operatorname{ber}_{-\frac{1}{2}}\left(x\sqrt{2}\right)=\frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}\left(e^{x}\sin\left(x+\frac{\pi}{8}\right)-e^{-x}\sin\left(x-\frac{\pi}{8}\right)\right)}}$ \Kelvinber{-\frac{1}{2}}@{x\sqrt{2}} = \frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}\left(e^{x}\sin@{x+\frac{\pi}{8}}-e^{-x}\sin@{x-\frac{\pi}{8}}\right)	${\displaystyle{\displaystyle\Re((-\frac{1}{2})+k+1)>0}}$	KelvinBer(-(1)/(2), x*sqrt(2)) = ((2)^(-(3)/(4)))/(sqrt(Pi*x))*(exp(x)*sin(x +(Pi)/(8))- exp(- x)*sin(x -(Pi)/(8)))	KelvinBer[-Divide[1,2], x*Sqrt[2]] == Divide[(2)^(-Divide[3,4]),Sqrt[Pi*x]]*(Exp[x]*Sin[x +Divide[Pi,8]]- Exp[- x]*Sin[x -Divide[Pi,8]])	Failure	Successful	Successful [Tested: 3]	Successful [Tested: 3]
10.61#Ex12	${\displaystyle{\displaystyle\operatorname{bei}_{-\frac{1}{2}}\left(x\sqrt{2}\right)=-\frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}\left(e^{x}\cos\left(x+\frac{\pi}{8}\right)+e^{-x}\cos\left(x-\frac{\pi}{8}\right)\right)}}$ \Kelvinbei{-\frac{1}{2}}@{x\sqrt{2}} = -\frac{2^{-\frac{3}{4}}}{\sqrt{\pi x}}\left(e^{x}\cos@{x+\frac{\pi}{8}}+e^{-x}\cos@{x-\frac{\pi}{8}}\right)		KelvinBei(-(1)/(2), x*sqrt(2)) = -((2)^(-(3)/(4)))/(sqrt(Pi*x))*(exp(x)*cos(x +(Pi)/(8))+ exp(- x)*cos(x -(Pi)/(8)))	KelvinBei[-Divide[1,2], x*Sqrt[2]] == -Divide[(2)^(-Divide[3,4]),Sqrt[Pi*x]]*(Exp[x]*Cos[x +Divide[Pi,8]]+ Exp[- x]*Cos[x -Divide[Pi,8]])	Failure	Successful	Successful [Tested: 3]	Successful [Tested: 3]
10.61.E11	${\displaystyle{\displaystyle\operatorname{ker}_{\frac{1}{2}}\left(x\sqrt{2}\right)=\operatorname{kei}_{-\frac{1}{2}}\left(x\sqrt{2}\right)}}$ \Kelvinker{\frac{1}{2}}@{x\sqrt{2}} = \Kelvinkei{-\frac{1}{2}}@{x\sqrt{2}}		KelvinKer((1)/(2), x*sqrt(2)) = KelvinKei(-(1)/(2), x*sqrt(2))	KelvinKer[Divide[1,2], x*Sqrt[2]] == KelvinKei[-Divide[1,2], x*Sqrt[2]]	Successful	Successful	Skip - symbolical successful subtest	Successful [Tested: 3]
10.61.E11	${\displaystyle{\displaystyle\operatorname{kei}_{-\frac{1}{2}}\left(x\sqrt{2}\right)=-2^{-\frac{3}{4}}\sqrt{\frac{\pi}{x}}e^{-x}\sin\left(x-\frac{\pi}{8}\right)}}$ \Kelvinkei{-\frac{1}{2}}@{x\sqrt{2}} = -2^{-\frac{3}{4}}\sqrt{\frac{\pi}{x}}e^{-x}\sin@{x-\frac{\pi}{8}}		KelvinKei(-(1)/(2), x*sqrt(2)) = - (2)^(-(3)/(4))*sqrt((Pi)/(x))*exp(- x)*sin(x -(Pi)/(8))	KelvinKei[-Divide[1,2], x*Sqrt[2]] == - (2)^(-Divide[3,4])*Sqrt[Divide[Pi,x]]*Exp[- x]*Sin[x -Divide[Pi,8]]	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
10.61.E12	${\displaystyle{\displaystyle\operatorname{kei}_{\frac{1}{2}}\left(x\sqrt{2}\right)=-\operatorname{ker}_{-\frac{1}{2}}\left(x\sqrt{2}\right)}}$ \Kelvinkei{\frac{1}{2}}@{x\sqrt{2}} = -\Kelvinker{-\frac{1}{2}}@{x\sqrt{2}}		KelvinKei((1)/(2), x*sqrt(2)) = - KelvinKer(-(1)/(2), x*sqrt(2))	KelvinKei[Divide[1,2], x*Sqrt[2]] == - KelvinKer[-Divide[1,2], x*Sqrt[2]]	Successful	Successful	Skip - symbolical successful subtest	Successful [Tested: 3]
10.61.E12	${\displaystyle{\displaystyle-\operatorname{ker}_{-\frac{1}{2}}\left(x\sqrt{2}\right)=-2^{-\frac{3}{4}}\sqrt{\frac{\pi}{x}}e^{-x}\cos\left(x-\frac{\pi}{8}\right)}}$ -\Kelvinker{-\frac{1}{2}}@{x\sqrt{2}} = -2^{-\frac{3}{4}}\sqrt{\frac{\pi}{x}}e^{-x}\cos@{x-\frac{\pi}{8}}		- KelvinKer(-(1)/(2), x*sqrt(2)) = - (2)^(-(3)/(4))*sqrt((Pi)/(x))*exp(- x)*cos(x -(Pi)/(8))	- KelvinKer[-Divide[1,2], x*Sqrt[2]] == - (2)^(-Divide[3,4])*Sqrt[Divide[Pi,x]]*Exp[- x]*Cos[x -Divide[Pi,8]]	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]