In[]:=
$Version
Out[]=
14.2.0 for Microsoft Windows (64-bit) (December 26, 2024)
In[]:=
d[v_]:=HankelH1[v,k1b]+HankelH2[v,k1b];
In[]:=
D[d[v],v]//FunctionExpand
Out[]=
BesselJ[v,bk1]
v(-1+
2
v
)
-
vBesselJ[v,bk1]
-1+
2
v
+
-2v
2
2v
(bk1)
πBesselY[v,bk1]HypergeometricPFQv,
1
2
+v,{1+v,1+v,1+2v},-
2
b
2
k1

2
Gamma[1+v]
+
-2v
2
2v
(bk1)
πBesselJ[v,bk1]Cot[πv]HypergeometricPFQv,
1
2
+v,{1+v,1+v,1+2v},-
2
b
2
k1

(-1+
2
v
)
2
Gamma[1+v]
-
-2v
2
2v
(bk1)
π
2
v
BesselJ[v,bk1]Cot[πv]HypergeometricPFQv,
1
2
+v,{1+v,1+v,1+2v},-
2
b
2
k1

(-1+
2
v
)
2
Gamma[1+v]
+
2
b
2
k1
BesselJ[v,bk1]HypergeometricPFQ1,1,
3
2
,{2,2,2-v,2+v},-
2
b
2
k1

2(-1+
2
v
)
-2BesselJ[v,bk1]Log[2]-
2BesselJ[v,bk1]Log[bk1]
-1+
2
v
+
2
2
v
BesselJ[v,bk1]Log[bk1]
-1+
2
v
-2BesselJ[v,bk1]PolyGamma[0,v]
​