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Miklós Schweitzer
2011 Miklós Schweitzer
8
8
Part of
2011 Miklós Schweitzer
Problems
(1)
real analysis
Source: miklos schweitzer 2011 q8
8/29/2021
Given a nonzero real number
a
≤
1
/
e
a\leq 1/e
a
≤
1/
e
, let
z
1
,
.
.
.
,
z
n
∈
C
z_1, ..., z_n \in C
z
1
,
...
,
z
n
∈
C
be non-real numbers for which
z
e
z
+
a
=
0
ze^z + a = 0
z
e
z
+
a
=
0
holds, and let
c
1
,
.
.
.
,
c
n
∈
C
c_1, ..., c_n \in C
c
1
,
...
,
c
n
∈
C
be arbitrary. Show that the function
f
(
x
)
=
R
e
(
∑
j
=
1
n
c
j
e
z
j
x
)
f(x)=Re(\sum_{j=1}^n c_j e^{z_j x})
f
(
x
)
=
R
e
(
∑
j
=
1
n
c
j
e
z
j
x
)
(
x
∈
R
x \in R
x
∈
R
) has a zero in every closed interval of length 1.
real analysis