Harmonic conjugate
Imagine you have two friends, Jack and Jill, who like to play a game where they always stand opposite each other on a circular field. Jack will stand at point A on the edge of the circle and Jill will stand directly across from him, at point B. They both have a ball, and they begin to toss it back and forth.
Now, if someone were to ask you where the ball is at any given moment, you could simply say that it's either with Jack at point A or with Jill at point B. There's no ambiguity about it.
But what if Jack and Jill could do something very special. They could each create a new imaginary friend, whom they could control and who would borrow the ball from them midway through their toss, then pass it on to them after a moment, and then vanish. Jack's imaginary friend would always be standing at point C, halfway between Jack and Jill, and Jill's imaginary friend would always be at point D, also halfway between Jack and Jill.
Now, when the ball is in the hands of one of the imaginary friends, it's not really at any particular point that we can see. But we could still keep track of it by knowing which imaginary friend has it at the moment. We would refer to the ball as being "with Jack's imaginary friend" or "with Jill's imaginary friend."
Notice how unusual it is that we can keep track of something that doesn't really exist, just by knowing who has control of it at any given moment. That's the key idea behind what mathematicians call "harmonic conjugates."
Mathematically speaking, a harmonic conjugate is a function that's related to another function in a particular way. Let's say we have a function that describes the height of a hill at each point on a map. The harmonic conjugate of this function is another function that describes the slope of the hill at each point.
Now, the slope of the hill is not something we can directly observe. But we can still define it as a function in terms of the height function. Just as with the imaginary friends who controlled the ball, the slope function "borrows" the height function for a moment, does some calculations, and returns a result that tells us something about the slope.
So even though we can't see or touch the slope itself, we can still compute it using the height function and its harmonic conjugate function. The two functions are said to be "harmonically related" to each other in the same way that each imaginary friend was related to the teammate who launched the ball from the opposite side of the circular field.
Now, if someone were to ask you where the ball is at any given moment, you could simply say that it's either with Jack at point A or with Jill at point B. There's no ambiguity about it.
But what if Jack and Jill could do something very special. They could each create a new imaginary friend, whom they could control and who would borrow the ball from them midway through their toss, then pass it on to them after a moment, and then vanish. Jack's imaginary friend would always be standing at point C, halfway between Jack and Jill, and Jill's imaginary friend would always be at point D, also halfway between Jack and Jill.
Now, when the ball is in the hands of one of the imaginary friends, it's not really at any particular point that we can see. But we could still keep track of it by knowing which imaginary friend has it at the moment. We would refer to the ball as being "with Jack's imaginary friend" or "with Jill's imaginary friend."
Notice how unusual it is that we can keep track of something that doesn't really exist, just by knowing who has control of it at any given moment. That's the key idea behind what mathematicians call "harmonic conjugates."
Mathematically speaking, a harmonic conjugate is a function that's related to another function in a particular way. Let's say we have a function that describes the height of a hill at each point on a map. The harmonic conjugate of this function is another function that describes the slope of the hill at each point.
Now, the slope of the hill is not something we can directly observe. But we can still define it as a function in terms of the height function. Just as with the imaginary friends who controlled the ball, the slope function "borrows" the height function for a moment, does some calculations, and returns a result that tells us something about the slope.
So even though we can't see or touch the slope itself, we can still compute it using the height function and its harmonic conjugate function. The two functions are said to be "harmonically related" to each other in the same way that each imaginary friend was related to the teammate who launched the ball from the opposite side of the circular field.