lim z z frac f z f z z z math superficially definition formally analogous derivative real function however complex derivatives differentiable functions behave significantly different ways compared real counterparts particular limit exist value difference quotient must approach complex number regardless manner approach math z math complex plane consequently complex differentiability much stronger implications real differentiability instance holomorphic functions infinitely differentiable whereas existence ''n''th derivative need imply existence
Formal science Mathematics Pure mathematics Analysis Complex analysis