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Lemma: If a harmonic function $h$ vanishes on two lines $L₁$ and $L₂$ and if the angle between $L₁$ and $L₂$ is not a rational multiple of $π$. Then $h$ vanishes identically.

The proof of this result can be found in: Harmonic functions with zeros on two lines

My question is:

Find the contrapositive of this lemma.

I want some thing like this: The function $h$ is not vanishes identically imply that $h$ vanishes on two lines $L₁$ and $L₂$ and the angle between $L₁$ and $L₂$ is a rational multiple of $π$

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John2000
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1 Answers1

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The lemma is of the form

Suppose X. If A, then B.

There are two different statements that might be called the converse, depending on context. First:

Suppose X. If B, then A.

Second:

If $B$, then $A$ and $X$.

But it seems from the question that you are actually looking for the contrapositive, not the converse. The converse will not negate the conclusion - in this case, the converse will assume $h$ does vanish identically. In general the converse is the additional statement needed to turn an "only if" into an "if and only if".

For the contrapositive, based onthe form of the lemma, the two possibilities are

Suppose X. If not B then not A.

and

If not B then (not A or not X).

The first of those is more likely to be useful here. It would say

Suppose $h$ is a harmonic function that vanishes on two lines $L_1$ and $L_2$. If $h$ does not vanish identically, then the angle between the two lines is a rational multiple of $\pi$.

By the way, the reason that there are multiple options is that the original statement (like many) is not the most simple kind of "if/then"; it has a nested structure.

Carl Mummert
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