Basic Idea
A function f(z) defined on the complex plane is said to be of exponential type if there exist constants M and τ such that
in the limit of . Here, the complex variable z was written as to emphasize that the limit must hold in all directions θ. Letting τ stand for the infimum of all such τ, one then says that the function f is of exponential type τ.
For example, let . Then one says that is of exponential type π, since π is the smallest number that bounds the growth of along the imaginary axis. So, for this example, Carlson's theorem cannot apply, as it requires functions of exponential type less than π. Similarly, the Euler-MacLaurin formula cannot be applied either, as it, too, expresses an theorem ultimately anchored in the theory of finite differences.
Read more about this topic: Exponential Type
Famous quotes containing the words basic idea, basic and/or idea:
“Our basic ideas about how to parent are encrusted with deeply felt emotions and many myths. One of the myths of parenting is that it is always fun and games, joy and delight. Everyone who has been a parent will testify that it is also anxiety, strife, frustration, and even hostility. Thus most major parenting- education formats deal with parental emotions and attitudes and, to a greater or lesser extent, advocate that the emotional component is more important than the knowledge.”
—Bettye M. Caldwell (20th century)
“When you realize how hard it is to know the truth about yourself, you understand that even the most exhaustive and well-meaning autobiography, determined to tell the truth, represents, at best, a guess. There have been times in my life when I felt incredibly happy. Life was full. I seemed productive. Then I thought,”Am I really happy or am I merely masking a deep depression with frantic activity?” If I don’t know such basic things about myself, who does?”
—Phyllis Rose (b. 1942)
“The idea of enemies is awful it makes one stop remembering eternity and the fear of death. That is what enemies are. Possessions are the same as enemies only less so, they too make one forget eternity and the fear of death.”
—Gertrude Stein (1874–1946)