In[]:=
Ein[z_]:=ExpIntegralE[1,z]+Log[z]+EulerGamma;
In[]:=
ein[z_]=Normal@FullSimplify@Series[Ein[z],{z,-1,10}]
Out[]=
EulerGamma+(-1+)(1+z)-
1
2
2
(1+z)
+
1
6
(-2+)
3
(1+z)
+
1
12
(-3+)
4
(1+z)
+
1
40
(-8+3)
5
(1+z)
+-
1
6
+
11
180
6
(1+z)
+-
1
7
+
53
1008
7
(1+z)
+-
1
8
+
103
2240
8
(1+z)
+-
1
9
+
2119
51840
9
(1+z)
+-
1
10
+
16687
453600
10
(1+z)
-ExpIntegralEi[1]
!(Arg[1+z]<0⊻Im[z]<0)
EulerGamma+2π+(-1+)(1+z)-
1
2
2
(1+z)
+
1
6
(-2+)
3
(1+z)
+
1
12
(-3+)
4
(1+z)
+
1
40
(-8+3)
5
(1+z)
+-
1
6
+
11
180
6
(1+z)
+-
1
7
+
53
1008
7
(1+z)
+-
1
8
+
103
2240
8
(1+z)
+-
1
9
+
2119
51840
9
(1+z)
+-
1
10
+
16687
453600
10
(1+z)
-ExpIntegralEi[1]
Im[z]≥0&&Arg[1+z]<0
EulerGamma-2π+(-1+)(1+z)-
1
2
2
(1+z)
+
1
6
(-2+)
3
(1+z)
+
1
12
(-3+)
4
(1+z)
+
1
40
(-8+3)
5
(1+z)
+-
1
6
+
11
180
6
(1+z)
+-
1
7
+
53
1008
7
(1+z)
+-
1
8
+
103
2240
8
(1+z)
+-
1
9
+
2119
51840
9
(1+z)
+-
1
10
+
16687
453600
10
(1+z)
-ExpIntegralEi[1]
True
In[]:=
ComplexPlot3D[ein[z],{z,-1-I,1+I}]
Out[]=
In[]:=
ComplexPlot3D[Ein[z],{z,-1-I,1+I}]
Out[]=